id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-17300 | Superset of Dependent Set is Dependent/Corollary | Let $A \subseteq S$.
Let $x \in A$.
If $x$ is a loop then $A$ is dependent. | Let $x$ be a loop.
By definition of a loop:
:$\set x \notin \mathscr I$
By definition of a dependent subset:
:$\set x$ is a dependent subset
From Singleton of Element is Subset:
:$\set x \subseteq A$
From Superset of Dependent Set is Dependent:
:$A$ is a dependent subset
{{qed}} | Let $A \subseteq S$.
Let $x \in A$.
If $x$ is a [[Definition:Loop (Matroid)|loop]] then $A$ is [[Definition:Dependent Subset (Matroid)|dependent]]. | Let $x$ be a [[Definition:Loop (Matroid)|loop]].
By definition of a [[Definition:Loop (Matroid)|loop]]:
:$\set x \notin \mathscr I$
By definition of a [[Definition:Dependent Subset (Matroid)|dependent subset]]:
:$\set x$ is a [[Definition:Dependent Subset (Matroid)|dependent subset]]
From [[Singleton of Element is S... | Superset of Dependent Set is Dependent/Corollary | https://proofwiki.org/wiki/Superset_of_Dependent_Set_is_Dependent/Corollary | https://proofwiki.org/wiki/Superset_of_Dependent_Set_is_Dependent/Corollary | [
"Matroid Dependent Subsets",
"Matroid Loops"
] | [
"Definition:Loop (Matroid)",
"Definition:Matroid/Dependent Set"
] | [
"Definition:Loop (Matroid)",
"Definition:Loop (Matroid)",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Dependent Set",
"Singleton of Element is Subset",
"Superset of Dependent Set is Dependent",
"Definition:Matroid/Dependent Set"
] |
proofwiki-17301 | Closure of Subset contains Loop | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $x$ be a loop of $M$.
Let $A \subseteq S$.
Then:
:$x \in \map \sigma A$
where $\map \sigma A$ denotes the closure of $A$. | By definition of the closure of $A$:
:$x \in \map \sigma A$ {{iff}} $x \sim A$
where $\sim$ is the depends relation on $M$.
By definition of the depends relation:
:$x \sim A$ {{iff}} $\map \rho {A \cup \set x} = \map \rho A$
where $\rho$ is the rank function on $M$.
So it remains to show that:
:$\map \rho {A \cup \set ... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $x$ be a [[Definition:Loop (Matroid)|loop]] of $M$.
Let $A \subseteq S$.
Then:
:$x \in \map \sigma A$
where $\map \sigma A$ denotes the [[Definition:Closure Operator (Matroid)|closure]] of $A$. | By definition of the [[Definition:Closure Operator (Matroid)|closure]] of $A$:
:$x \in \map \sigma A$ {{iff}} $x \sim A$
where $\sim$ is the [[Definition:Depends Relation (Matroid)|depends relation]] on $M$.
By definition of the [[Definition:Depends Relation (Matroid)|depends relation]]:
:$x \sim A$ {{iff}} $\map \rho... | Closure of Subset contains Loop | https://proofwiki.org/wiki/Closure_of_Subset_contains_Loop | https://proofwiki.org/wiki/Closure_of_Subset_contains_Loop | [
"Matroid Closure",
"Matroid Loops"
] | [
"Definition:Matroid",
"Definition:Loop (Matroid)",
"Definition:Closure Operator (Matroid)"
] | [
"Definition:Closure Operator (Matroid)",
"Definition:Depends Relation (Matroid)",
"Definition:Depends Relation (Matroid)",
"Definition:Rank Function (Matroid)",
"Definition:Rank Function (Matroid)",
"Max Operation Equals an Operand",
"Definition:Contrapositive Statement",
"Superset of Dependent Set is... |
proofwiki-17302 | Element is Loop iff Singleton is Circuit | Let $M = \struct{S, \mathscr I}$ be a matroid.
Let $x \in S$.
Then:
:$x$ is a loop {{iff}} $\set x$ is a circuit | === Necessary Condition ===
Let $x$ be a loop.
By definition of a loop:
:$\set x$ is a dependent subset of $S$
Let $A \subseteq \set x$ be a dependent subset.
From Power Set of Singleton:
:$\powerset {\set x} = \set{\O, \set x}$
By matroid axiom $(\text I 1)$:
:$\O$ is an independent subset
Then:
:$A = \set x$
It follo... | Let $M = \struct{S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $x \in S$.
Then:
:$x$ is a [[Definition:Loop (Matroid)|loop]] {{iff}} $\set x$ is a [[Definition:Circuit (Matroid)|circuit]] | === Necessary Condition ===
Let $x$ be a [[Definition:Loop (Matroid)|loop]].
By definition of a [[Definition:Loop (Matroid)|loop]]:
:$\set x$ is a [[Definition:Dependent Subset (Matroid)|dependent subset]] of $S$
Let $A \subseteq \set x$ be a [[Definition:Dependent Subset (Matroid)|dependent subset]].
From [[Power ... | Element is Loop iff Singleton is Circuit | https://proofwiki.org/wiki/Element_is_Loop_iff_Singleton_is_Circuit | https://proofwiki.org/wiki/Element_is_Loop_iff_Singleton_is_Circuit | [
"Matroid Loops",
"Matroid Circuits"
] | [
"Definition:Matroid",
"Definition:Loop (Matroid)",
"Definition:Circuit (Matroid)"
] | [
"Definition:Loop (Matroid)",
"Definition:Loop (Matroid)",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Dependent Set",
"Power Set of Singleton",
"Axiom:Matroid Axioms",
"Definition:Matroid/Independent Set",
"Definition:Minimal",
"Definition:Matroid/Dependent Set",
"Definition:Circuit (M... |
proofwiki-17303 | Element is Member of Base iff Not Loop | Let $M = \struct{S, \mathscr I}$ be a matroid.
Let $\mathscr B$ denote the set of all bases of $M$.
Let $x \in S$.
Then:
:$\exists B \in \mathscr B: x \in B$ {{iff}} $x$ is not a loop | === Necessary Condition ===
Let $B \in \mathscr B$ such that $x \in B$.
From Singleton of Element is Subset:
:$\set x \subseteq B$
By definition of a base:
:$B \in \mathscr I$
From matroid axiom $(\text I 2)$:
:$\set x \in \mathscr I$
Then $\set x$ is not a dependent subset by definition.
It follows that $x$ is not a l... | Let $M = \struct{S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\mathscr B$ denote the set of all [[Definition:Base of Matroid|bases]] of $M$.
Let $x \in S$.
Then:
:$\exists B \in \mathscr B: x \in B$ {{iff}} $x$ is not a [[Definition:Loop (Matroid)|loop]] | === Necessary Condition ===
Let $B \in \mathscr B$ such that $x \in B$.
From [[Singleton of Element is Subset]]:
:$\set x \subseteq B$
By definition of a [[Definition:Base of Matroid|base]]:
:$B \in \mathscr I$
From [[Axiom:Matroid Axioms|matroid axiom $(\text I 2)$]]:
:$\set x \in \mathscr I$
Then $\set x$ is n... | Element is Member of Base iff Not Loop | https://proofwiki.org/wiki/Element_is_Member_of_Base_iff_Not_Loop | https://proofwiki.org/wiki/Element_is_Member_of_Base_iff_Not_Loop | [
"Matroid Bases",
"Matroid Loops"
] | [
"Definition:Matroid",
"Definition:Base of Matroid",
"Definition:Loop (Matroid)"
] | [
"Singleton of Element is Subset",
"Definition:Base of Matroid",
"Axiom:Matroid Axioms",
"Definition:Matroid/Dependent Set",
"Definition:Loop",
"Definition:Loop",
"Definition:Loop",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Dependent Set"
] |
proofwiki-17304 | Distinct Elements are Parallel iff Pair forms Circuit | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $x, y \in S : x \ne y$.
Then:
:$x$ and $y$ are parallel {{iff}} $\set {x, y}$ is a circuit | === Necessary Condition ===
Let $x$ and $y$ be parallel.
By definition of parallel:
:$\set x$ is independent
:$\set y$ is independent
:$\set {x, y}$ is dependent
Let $A \subseteq \set {x, y}$ be dependent.
Thus:
:$A \ne \set x, \set y$
By matroid axiom $(\text I 1)$:
:$\O$ is independent
Thus:
:$A \ne \O$
From Power ... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $x, y \in S : x \ne y$.
Then:
:$x$ and $y$ are [[Definition:Parallel (Matroid)|parallel]] {{iff}} $\set {x, y}$ is a [[Definition:Circuit (Matroid)|circuit]] | === Necessary Condition ===
Let $x$ and $y$ be [[Definition:Parallel (Matroid)|parallel]].
By definition of [[Definition:Parallel (Matroid)|parallel]]:
:$\set x$ is [[Definition:Independent Subset (Matroid)|independent]]
:$\set y$ is [[Definition:Independent Subset (Matroid)|independent]]
:$\set {x, y}$ is [[Defini... | Distinct Elements are Parallel iff Pair forms Circuit | https://proofwiki.org/wiki/Distinct_Elements_are_Parallel_iff_Pair_forms_Circuit | https://proofwiki.org/wiki/Distinct_Elements_are_Parallel_iff_Pair_forms_Circuit | [
"Matroid Parallelism",
"Matroid Circuits"
] | [
"Definition:Matroid",
"Definition:Parallel (Matroid)",
"Definition:Circuit (Matroid)"
] | [
"Definition:Parallel (Matroid)",
"Definition:Parallel (Matroid)",
"Definition:Matroid/Independent Set",
"Definition:Matroid/Independent Set",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Dependent Set",
"Axiom:Matroid Axioms",
"Definition:Matroid/Independent Set",
"Power Set of Doubleton"... |
proofwiki-17305 | Parallel Relationship is Transitive | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $x, y, z \in S : x \ne y, x \ne z, y \ne z$.
If $x$ is parallel to $y$ and $y$ is parallel to $z$ then $x$ is parallel to $z$. | Let $x$ be parallel to $y$ and $y$ be parallel to $z$.
By definition of parallel:
:$\set x$, $\set y$, $\set z$ are independent subsets
:$\set {x, y}$, $\set {y, z}$ are dependent subsets
To show that $x$ is parallel to $z$ it remains to show that:
:$\set {x, z}$ is dependent
{{AimForCont}} $\set {x, z}$ is independent... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $x, y, z \in S : x \ne y, x \ne z, y \ne z$.
If $x$ is [[Definition:Parallel (Matroid)|parallel]] to $y$ and $y$ is [[Definition:Parallel (Matroid)|parallel]] to $z$ then $x$ is [[Definition:Parallel (Matroid)|parallel]] to $z$. | Let $x$ be [[Definition:Parallel (Matroid)|parallel]] to $y$ and $y$ be [[Definition:Parallel (Matroid)|parallel]] to $z$.
By definition of [[Definition:Parallel (Matroid)|parallel]]:
:$\set x$, $\set y$, $\set z$ are [[Definition:Independent Subset (Matroid)|independent subsets]]
:$\set {x, y}$, $\set {y, z}$ are [[D... | Parallel Relationship is Transitive | https://proofwiki.org/wiki/Parallel_Relationship_is_Transitive | https://proofwiki.org/wiki/Parallel_Relationship_is_Transitive | [
"Matroid Parallelism"
] | [
"Definition:Matroid",
"Definition:Parallel (Matroid)",
"Definition:Parallel (Matroid)",
"Definition:Parallel (Matroid)"
] | [
"Definition:Parallel (Matroid)",
"Definition:Parallel (Matroid)",
"Definition:Parallel (Matroid)",
"Definition:Matroid/Independent Set",
"Definition:Matroid/Dependent Set",
"Definition:Parallel (Matroid)",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Independent Set",
"Axiom:Matroid Axiom... |
proofwiki-17306 | Distinct Matroid Elements are Parallel iff Each is in Closure of Other | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\sigma: \powerset S \to \powerset S$ denote the closure operator of $M$.
Let $x, y \in S : x \ne y$.
Then $x$ is parallel to $y$ {{iff}}:
:$(1)\quad x$ and $y$ are not loops
:$(2)\quad x \in \map \sigma {\set y}$
:$(3)\quad y \in \map \sigma {\set x}$ | === Lemma ===
{{:Distinct Matroid Elements are Parallel iff Each is in Closure of Other/Lemma}}{{qed|lemma}} | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\sigma: \powerset S \to \powerset S$ denote the [[Definition:Closure Operator (Matroid)|closure operator]] of $M$.
Let $x, y \in S : x \ne y$.
Then $x$ is [[Definition:Parallel (Matroid)|parallel]] to $y$ {{iff}}:
:$(1)\quad x$ and $y$ are ... | === [[Distinct Matroid Elements are Parallel iff Each is in Closure of Other/Lemma|Lemma]] ===
{{:Distinct Matroid Elements are Parallel iff Each is in Closure of Other/Lemma}}{{qed|lemma}} | Distinct Matroid Elements are Parallel iff Each is in Closure of Other | https://proofwiki.org/wiki/Distinct_Matroid_Elements_are_Parallel_iff_Each_is_in_Closure_of_Other | https://proofwiki.org/wiki/Distinct_Matroid_Elements_are_Parallel_iff_Each_is_in_Closure_of_Other | [
"Matroid Closure",
"Matroid Parallelism",
"Distinct Matroid Elements are Parallel iff Each is in Closure of Other"
] | [
"Definition:Matroid",
"Definition:Closure Operator (Matroid)",
"Definition:Parallel (Matroid)",
"Definition:Loop (Matroid)"
] | [
"Distinct Matroid Elements are Parallel iff Each is in Closure of Other/Lemma"
] |
proofwiki-17307 | Closure of Subset Contains Parallel Elements | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\sigma: \powerset S \to \powerset S$ denote the closure operator of $M$.
Let $A \subseteq S$.
Let $x, y \in S$.
If $x \in \map \sigma A$ and $y$ is parallel to $x$ then:
:$y \in \map \sigma A$ | Let $\rho: \powerset S \to \Z$ denote the rank function of $M$.
Let $x \in \map \sigma A$
Let $y$ be parallel to $x$.
By the definitions of the closure operator and depends depends:
:$\map \rho A = \map \rho {A \cup \set x}$
and:
:$y \in \map \sigma A$ {{iff}} $\map \rho A = \map \rho {A \cup \set y}$
From Rank Functio... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\sigma: \powerset S \to \powerset S$ denote the [[Definition:Closure Operator (Matroid)|closure operator]] of $M$.
Let $A \subseteq S$.
Let $x, y \in S$.
If $x \in \map \sigma A$ and $y$ is [[Definition:Parallel (Matroid)|parallel]] to $x$... | Let $\rho: \powerset S \to \Z$ denote the [[Definition:Rank Function (Matroid)|rank function]] of $M$.
Let $x \in \map \sigma A$
Let $y$ be [[Definition:Parallel (Matroid)|parallel]] to $x$.
By the definitions of the [[Definition:Closure Operator (Matroid)|closure operator]] and [[Definition:Depends Relation (Matr... | Closure of Subset Contains Parallel Elements | https://proofwiki.org/wiki/Closure_of_Subset_Contains_Parallel_Elements | https://proofwiki.org/wiki/Closure_of_Subset_Contains_Parallel_Elements | [
"Matroid Closure",
"Matroid Parallelism"
] | [
"Definition:Matroid",
"Definition:Closure Operator (Matroid)",
"Definition:Parallel (Matroid)"
] | [
"Definition:Rank Function (Matroid)",
"Definition:Parallel (Matroid)",
"Definition:Closure Operator (Matroid)",
"Definition:Depends Relation (Matroid)",
"Rank Function is Increasing",
"Rank Function is Increasing",
"Definition:Parallel (Matroid)",
"Definition:Matroid/Independent Set",
"Independent S... |
proofwiki-17308 | Set with Two Parallel Elements is Dependent | Let $M = \struct{S, \mathscr I}$ be a matroid.
Let $A \subseteq S$.
Let $x, y \in S$.
Let $x, y$ be parallel elements.
If $x, y \in A$ then $A$ is dependent. | Let $x, y \in A$.
From Doubleton of Elements is Subset:
:$\set{x, y} \subseteq A$
By the definition of parallel elements:
:$\set {x, y}$ is dependent
From Superset of Dependent Set is Dependent:
:$A$ is dependent
{{qed}} | Let $M = \struct{S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $A \subseteq S$.
Let $x, y \in S$.
Let $x, y$ be [[Definition:Parallel (Matroid)|parallel]] [[Definition:Element|elements]].
If $x, y \in A$ then $A$ is [[Definition:Dependent Subset (Matroid)|dependent]]. | Let $x, y \in A$.
From [[Doubleton of Elements is Subset]]:
:$\set{x, y} \subseteq A$
By the definition of [[Definition:Parallel (Matroid)|parallel]] [[Definition:Element|elements]]:
:$\set {x, y}$ is [[Definition:Dependent Subset (Matroid)|dependent]]
From [[Superset of Dependent Set is Dependent]]:
:$A$ is [[Defi... | Set with Two Parallel Elements is Dependent | https://proofwiki.org/wiki/Set_with_Two_Parallel_Elements_is_Dependent | https://proofwiki.org/wiki/Set_with_Two_Parallel_Elements_is_Dependent | [
"Matroid Parallelism",
"Matroid Dependent Subsets"
] | [
"Definition:Matroid",
"Definition:Parallel (Matroid)",
"Definition:Element",
"Definition:Matroid/Dependent Set"
] | [
"Doubleton of Elements is Subset",
"Definition:Parallel (Matroid)",
"Definition:Element",
"Definition:Matroid/Dependent Set",
"Superset of Dependent Set is Dependent",
"Definition:Matroid/Dependent Set"
] |
proofwiki-17309 | Loop Belongs to Every Flat | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $A \subseteq S$.
Let $x \in S$.
If $x$ is a loop and $A$ is a flat subset then $x \in A$. | Let $\rho: \powerset S \to \Z$ denote the rank function of $M$.
We proceed by Proof by Contraposition.
That is, it is shown that:
:if $x \notin A$ then either $x$ is not a loop or $A$ is not a flat subset
Let $x \notin A$.
Let $x$ be a loop.
By definition of a loop:
:$\set x$ is a dependent subset.
From Rank Function i... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $A \subseteq S$.
Let $x \in S$.
If $x$ is a [[Definition:Loop (Matroid)|loop]] and $A$ is a [[Definition:Flat (Matroid)|flat subset]] then $x \in A$. | Let $\rho: \powerset S \to \Z$ denote the [[Definition:Rank Function (Matroid)|rank function]] of $M$.
We proceed by [[Proof by Contraposition]].
That is, it is shown that:
:if $x \notin A$ then either $x$ is not a [[Definition:Loop (Matroid)|loop]] or $A$ is not a [[Definition:Flat (Matroid)|flat subset]]
Let $x \... | Loop Belongs to Every Flat | https://proofwiki.org/wiki/Loop_Belongs_to_Every_Flat | https://proofwiki.org/wiki/Loop_Belongs_to_Every_Flat | [
"Matroid Loops",
"Matroid Flats"
] | [
"Definition:Matroid",
"Definition:Loop (Matroid)",
"Definition:Flat (Matroid)"
] | [
"Definition:Rank Function (Matroid)",
"Proof by Contraposition",
"Definition:Loop (Matroid)",
"Definition:Flat (Matroid)",
"Definition:Loop (Matroid)",
"Definition:Loop (Matroid)",
"Definition:Matroid/Dependent Set",
"Rank Function is Increasing",
"Superset of Dependent Set is Dependent",
"Singlet... |
proofwiki-17310 | Parallel Elements Depend on Same Subsets | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $A \subseteq S$.
Let $x, y \in S$.
Let $x$ be parallel to $y$.
Then:
:$x$ depends on $A$ {{iff}} $y$ depends on $A$ | This follows directly from Closure of Subset Contains Parallel Elements.
{{qed}} | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $A \subseteq S$.
Let $x, y \in S$.
Let $x$ be [[Definition:Parallel (Matroid)|parallel]] to $y$.
Then:
:$x$ [[Definition:Depends Relation (Matroid)|depends]] on $A$ {{iff}} $y$ [[Definition:Depends Relation (Matroid)|depends]] on $A$ | This follows directly from [[Closure of Subset Contains Parallel Elements]].
{{qed}} | Parallel Elements Depend on Same Subsets | https://proofwiki.org/wiki/Parallel_Elements_Depend_on_Same_Subsets | https://proofwiki.org/wiki/Parallel_Elements_Depend_on_Same_Subsets | [
"Matroid Parallelism",
"Matroid Dependence"
] | [
"Definition:Matroid",
"Definition:Parallel (Matroid)",
"Definition:Depends Relation (Matroid)",
"Definition:Depends Relation (Matroid)"
] | [
"Closure of Subset Contains Parallel Elements"
] |
proofwiki-17311 | Matroid Contains No Loops iff Empty Set is Flat | Let $M = \struct{S, \mathscr I}$ be a matroid.
Then:
:$M$ contains no loops {{iff}} the empty set is flat. | Let $\rho: \powerset S \to \Z$ denote the rank function of $M$. | Let $M = \struct{S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Then:
:$M$ contains no [[Definition:Loop (Matroid)|loops]] {{iff}} the [[Definition:Empty Set|empty set]] is [[Definition:Flat (Matroid)|flat]]. | Let $\rho: \powerset S \to \Z$ denote the [[Definition:Rank Function (Matroid)|rank function]] of $M$. | Matroid Contains No Loops iff Empty Set is Flat | https://proofwiki.org/wiki/Matroid_Contains_No_Loops_iff_Empty_Set_is_Flat | https://proofwiki.org/wiki/Matroid_Contains_No_Loops_iff_Empty_Set_is_Flat | [
"Matroid Loops",
"Matroid Flats"
] | [
"Definition:Matroid",
"Definition:Loop (Matroid)",
"Definition:Empty Set",
"Definition:Flat (Matroid)"
] | [
"Definition:Rank Function (Matroid)"
] |
proofwiki-17312 | Absolutely Continuous Real Function is Uniformly Continuous | Let $I \subseteq \R$ be a real interval.
Let $f: I \to \R$ be an absolutely continuous real function.
Then $f$ is uniformly continuous. | Let $\epsilon$ be a positive real number.
Since $f$ is absolutely continuous, there exists real $\delta > 0$ such that for all collections of pairwise disjoint closed real intervals $\closedint {a_1} {b_1}, \dotsc, \closedint {a_n} {b_n} \subseteq I$ with:
:$\ds \sum_{i \mathop = 1}^n \paren {b_i - a_i} < \delta$
we h... | Let $I \subseteq \R$ be a [[Definition:Real Interval|real interval]].
Let $f: I \to \R$ be an [[Definition:Absolutely Continuous Real Function|absolutely continuous real function]].
Then $f$ is [[Definition:Uniformly Continuous Real Function|uniformly continuous]]. | Let $\epsilon$ be a [[Definition:Positive Real Number|positive real number]].
Since $f$ is [[Definition:Absolutely Continuous Real Function|absolutely continuous]], there exists [[Definition:Real Number|real]] $\delta > 0$ such that for all collections of [[Definition:Pairwise Disjoint|pairwise disjoint]] [[Definition... | Absolutely Continuous Real Function is Uniformly Continuous | https://proofwiki.org/wiki/Absolutely_Continuous_Real_Function_is_Uniformly_Continuous | https://proofwiki.org/wiki/Absolutely_Continuous_Real_Function_is_Uniformly_Continuous | [
"Absolutely Continuous Real Functions",
"Uniformly Continuous Real Functions"
] | [
"Definition:Real Interval",
"Definition:Absolute Continuity/Real Function",
"Definition:Uniform Continuity/Real Function"
] | [
"Definition:Positive/Real Number",
"Definition:Absolute Continuity/Real Function",
"Definition:Real Number",
"Definition:Pairwise Disjoint",
"Definition:Real Interval/Closed",
"Definition:Absolute Continuity/Real Function",
"Definition:Uniform Continuity/Real Function",
"Category:Absolutely Continuous... |
proofwiki-17313 | Closure of Subspace of Normed Vector Space is Subspace | Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $Y \subseteq X$ be a subspace of $X$.
Let $Y^-$ be the closure of $Y$.
Then $Y^- \subseteq X$ is also a subspace of $X$. | Suppose $y \in Y^-$.
Then there is a sequence $\ds \sequence {y_n}_{n \mathop \in \N} \in Y$ which converges to $y$.
Suppose $y \in Y$ and $y$ is a limit point.
Then one can define a constant sequence:
:$\sequence {y_n}_{n \mathop \in \N} = y$ | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $Y \subseteq X$ be a [[Definition:Vector Subspace|subspace]] of $X$.
Let $Y^-$ be the [[Definition:Closure/Normed Vector Space|closure]] of $Y$.
Then $Y^- \subseteq X$ is also a [[Definition:Vector Subspace|subs... | Suppose $y \in Y^-$.
Then there is a [[Definition:Sequence|sequence]] $\ds \sequence {y_n}_{n \mathop \in \N} \in Y$ which [[Definition:Convergent Sequence in Normed Vector Space|converges]] to $y$.
Suppose $y \in Y$ and $y$ is a [[Definition:Limit Point (Normed Vector Space)|limit point]].
Then one can define a [[D... | Closure of Subspace of Normed Vector Space is Subspace | https://proofwiki.org/wiki/Closure_of_Subspace_of_Normed_Vector_Space_is_Subspace | https://proofwiki.org/wiki/Closure_of_Subspace_of_Normed_Vector_Space_is_Subspace | [
"Set Closures"
] | [
"Definition:Normed Vector Space",
"Definition:Vector Subspace",
"Definition:Closure/Normed Vector Space",
"Definition:Vector Subspace"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Limit Point/Normed Vector Space",
"Definition:Constant",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Limit Point/Normed Vector Space",
"Definition:Sequence",
"Definition:Limit Point/Normed Vector S... |
proofwiki-17314 | Closure of Convex Subset in Normed Vector Space is Convex | Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $C \subseteq X$ be a convex subset of $X$.
Let $C^-$ be the closure of $C$.
Then $C^- \subseteq X$ is also a convex subset of $X$. | Let $x, y \in C^-$.
Suppose $x, y$ are limit points.
Then there are sequences $\sequence {x_n}_{n \mathop \in \N}, \sequence {x_n}_{n \mathop \in \N}$ in $C$, such that:
:$\ds \lim_{n \mathop \to \infty} x_n = x$
:$\ds \lim_{n \mathop \to \infty} x_y = y$
Let $\alpha \in \closedint 0 1$.
Then:
{{begin-eqn}}
{{eqn | l =... | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $C \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Subset|subset]] of $X$.
Let $C^-$ be the [[Definition:Closure/Normed Vector Space|closure]] of $C$.
Then $C^- \subseteq X$ is als... | Let $x, y \in C^-$.
Suppose $x, y$ are [[Definition:Limit Point (Normed Vector Space)|limit points]].
Then there are [[Definition:Sequence|sequences]] $\sequence {x_n}_{n \mathop \in \N}, \sequence {x_n}_{n \mathop \in \N}$ in $C$, such that:
:$\ds \lim_{n \mathop \to \infty} x_n = x$
:$\ds \lim_{n \mathop \to \inf... | Closure of Convex Subset in Normed Vector Space is Convex | https://proofwiki.org/wiki/Closure_of_Convex_Subset_in_Normed_Vector_Space_is_Convex | https://proofwiki.org/wiki/Closure_of_Convex_Subset_in_Normed_Vector_Space_is_Convex | [
"Normed Vector Spaces",
"Set Closures",
"Convex Sets (Vector Spaces)",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Subset",
"Definition:Closure/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Subset"
] | [
"Definition:Limit Point/Normed Vector Space",
"Definition:Sequence",
"Definition:Convex Set (Vector Space)",
"Definition:Closed Set/Normed Vector Space",
"Definition:Limit Point/Normed Vector Space"
] |
proofwiki-17315 | Minimal Number of Distinct Prime Factors for Integer to have Abundancy Index Exceed Given Number | Let $r \in \R$.
Let $\mathbb P^-$ be the set of prime numbers with possibly finitely many numbers removed.
Define:
:$M = \min \set {m \in \N: \ds \prod_{i \mathop = 1}^m \frac {p_i} {p_i - 1} > r}$
where $p_i$ is the $i$th element of $\mathbb P^-$, ordered by size.
Then $M$ satisfies:
:$(1): \quad$ Every number formed ... | First we show that abundancy index is multiplicative.
Let $n \in \N$ and let $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ be its prime factorization.
Then the abundancy index of $n$ is:
{{begin-eqn}}
{{eqn | l = \frac {\map {\sigma_1} n} n
| r = \frac {\map {\sigma_1} {p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} } } {p_1^... | Let $r \in \R$.
Let $\mathbb P^-$ be the set of [[Definition:Prime Number|prime numbers]] with possibly [[Definition:Finitely Many|finitely many]] numbers removed.
Define:
:$M = \min \set {m \in \N: \ds \prod_{i \mathop = 1}^m \frac {p_i} {p_i - 1} > r}$
where $p_i$ is the $i$th element of $\mathbb P^-$, [[Definitio... | First we show that [[Definition:Abundancy Index|abundancy index]] is [[Definition:Multiplicative Arithmetic Function|multiplicative]].
Let $n \in \N$ and let $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ be its [[Definition:Prime Factorization|prime factorization]].
Then the [[Definition:Abundancy Index|abundancy index]... | Minimal Number of Distinct Prime Factors for Integer to have Abundancy Index Exceed Given Number | https://proofwiki.org/wiki/Minimal_Number_of_Distinct_Prime_Factors_for_Integer_to_have_Abundancy_Index_Exceed_Given_Number | https://proofwiki.org/wiki/Minimal_Number_of_Distinct_Prime_Factors_for_Integer_to_have_Abundancy_Index_Exceed_Given_Number | [
"Abundancy"
] | [
"Definition:Prime Number",
"Definition:Finite Set",
"Definition:Ordering on Natural Numbers",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Abundancy Index",
"Definition:Distinct",
"Definition:Prime Factor",
"Definition:Abundancy Index",
"Definition:Distinct",
"Definition:Prime F... | [
"Definition:Abundancy Index",
"Definition:Multiplicative Arithmetic Function",
"Definition:Prime Decomposition",
"Definition:Abundancy Index",
"Divisor Sum Function is Multiplicative",
"Definition:Abundancy Index",
"Definition:Multiplicative Arithmetic Function",
"Sum of Reciprocals of Primes is Diver... |
proofwiki-17316 | 1 plus Square is not Perfect Power | The equation:
:$x^p = y^2 + 1$
has no solution in the integers for $x, y, p > 1$. | Suppose $p$ is even.
Write $p = 2 k$.
Then:
{{begin-eqn}}
{{eqn | l = 1
| r = y^2 - x^{2 k}
}}
{{eqn | r = \paren {y - x^k} \paren {y + x^k}
| c = Difference of Two Squares
}}
{{end-eqn}}
Since both $y - x^k$ and $y + x^k$ are integers, they must be equal to $\pm 1$.
Summing them up, we have $2 y$ is one o... | The equation:
:$x^p = y^2 + 1$
has no solution in the [[Definition:Integer|integers]] for $x, y, p > 1$. | Suppose $p$ is [[Definition:Even Integer|even]].
Write $p = 2 k$.
Then:
{{begin-eqn}}
{{eqn | l = 1
| r = y^2 - x^{2 k}
}}
{{eqn | r = \paren {y - x^k} \paren {y + x^k}
| c = [[Difference of Two Squares]]
}}
{{end-eqn}}
Since both $y - x^k$ and $y + x^k$ are [[Definition:Integer|integers]], they must be... | 1 plus Square is not Perfect Power | https://proofwiki.org/wiki/1_plus_Square_is_not_Perfect_Power | https://proofwiki.org/wiki/1_plus_Square_is_not_Perfect_Power | [
"Number Theory"
] | [
"Definition:Integer"
] | [
"Definition:Even Integer",
"Difference of Two Squares",
"Definition:Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Even Integer",
"Odd Square Modulo 8",
"Definition:Contradiction",
"Definition:Even Integer",
"Definition:Odd Integer",
"Gauss... |
proofwiki-17317 | 1 plus Perfect Power is not Prime Power except for 9 | The only solution to:
:$x^m = y^n + 1$
is:
:$\tuple {x, m, y, n} = \tuple {3, 2, 2, 3}$
for positive integers $x, y, m, n > 1$, and $x$ is a prime number.
This is a special case of Catalan's Conjecture. | It suffices to show the result for prime values of $n$.
The case $n = 2$ is covered in 1 plus Square is not Perfect Power.
So we consider the cases where $n$ is an odd prime.
{{begin-eqn}}
{{eqn | l = x^m
| r = y^n + 1
}}
{{eqn | r = y^n - \paren {-1}^n
| c = as $n$ is odd
}}
{{eqn | r = \paren {y - \paren ... | The only solution to:
:$x^m = y^n + 1$
is:
:$\tuple {x, m, y, n} = \tuple {3, 2, 2, 3}$
for [[Definition:Positive Integer|positive integers]] $x, y, m, n > 1$, and $x$ is a [[Definition:Prime Number|prime number]].
This is a special case of [[Catalan's Conjecture]]. | It suffices to show the result for [[Definition:Prime Number|prime]] values of $n$.
The case $n = 2$ is covered in [[1 plus Square is not Perfect Power]].
So we consider the cases where $n$ is an [[Definition:Odd Integer|odd]] [[Definition:Prime Number|prime]].
{{begin-eqn}}
{{eqn | l = x^m
| r = y^n + 1
}}
{... | 1 plus Perfect Power is not Prime Power except for 9 | https://proofwiki.org/wiki/1_plus_Perfect_Power_is_not_Prime_Power_except_for_9 | https://proofwiki.org/wiki/1_plus_Perfect_Power_is_not_Prime_Power_except_for_9 | [
"Number Theory"
] | [
"Definition:Positive/Integer",
"Definition:Prime Number",
"Catalan's Conjecture"
] | [
"Definition:Prime Number",
"1 plus Square is not Perfect Power",
"Definition:Odd Integer",
"Definition:Prime Number",
"Definition:Odd Integer",
"Difference of Two Powers",
"Division Theorem for Polynomial Forms over Field",
"Definition:Odd Integer",
"Definition:Prime Number",
"Definition:Prime Num... |
proofwiki-17318 | Field is Principal Ideal Domain | Let $F$ be a field.
Then $F$ is a principal ideal domain. | Let $F$ be a field.
Let $I \subset F$ be a non-null ideal of $F$.
Let $a \in I$ be non-zero.
Since $F$ is a field, $a^{-1}$ exists.
We have that $1 = a^{-1} \cdot a \in I$.
Since $1 \in I$, for every element $b \in F$:
:$b = b \cdot 1 \in I$
we have that $I = F = \ideal 1$ if $I \ne \set 0$.
Thus the only ideals of $F$... | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Then $F$ is a [[Definition:Principal Ideal Domain|principal ideal domain]]. | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $I \subset F$ be a [[Definition:Non-Null Ideal|non-null]] [[Definition:Ideal of Ring|ideal]] of $F$.
Let $a \in I$ be non-[[Definition:Field Zero|zero]].
Since $F$ is a [[Definition:Field (Abstract Algebra)|field]], $a^{-1}$ exists.
We have that $1 = a... | Field is Principal Ideal Domain | https://proofwiki.org/wiki/Field_is_Principal_Ideal_Domain | https://proofwiki.org/wiki/Field_is_Principal_Ideal_Domain | [
"Principal Ideal Domains",
"Field Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Principal Ideal Domain"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Non-Null Ideal",
"Definition:Ideal of Ring",
"Definition:Field Zero",
"Definition:Field (Abstract Algebra)",
"Definition:Element",
"Definition:Ideal of Ring",
"Definition:Principal Ideal Domain",
"Definition:Principal Ideal Domain",
"Category:Prin... |
proofwiki-17319 | Complex Vector Space is Vector Space | Let $\C$ denote the set of complex numbers.
Then the complex vector space $\C^n$ is a vector space. | === Construction of Complex Vector Space ===
From the definition, a vector space is a unitary module whose scalar ring is a field.
In order to call attention to the precise scope of the operators, let complex addition and complex multiplication be expressed as $+_\C$ and $\times_\C$ respectively.
Then we can express th... | Let $\C$ denote the set of [[Definition:Complex Number|complex numbers]].
Then the [[Definition:Complex Vector Space|complex vector space $\C^n$]] is a [[Definition:Vector Space|vector space]]. | === Construction of Complex Vector Space ===
From the definition, a [[Definition:Vector Space|vector space]] is a [[Definition:Unitary Module|unitary module]] whose [[Definition:Scalar Ring of Unitary Module|scalar ring]] is a [[Definition:Field (Abstract Algebra)|field]].
In order to call attention to the precise s... | Complex Vector Space is Vector Space | https://proofwiki.org/wiki/Complex_Vector_Space_is_Vector_Space | https://proofwiki.org/wiki/Complex_Vector_Space_is_Vector_Space | [
"Examples of Vector Spaces",
"Complex Numbers"
] | [
"Definition:Complex Number",
"Definition:Complex Vector Space",
"Definition:Vector Space"
] | [
"Definition:Vector Space",
"Definition:Unitary Module over Ring",
"Definition:Scalar Ring/Unitary Module",
"Definition:Field (Abstract Algebra)",
"Definition:Addition/Complex Numbers",
"Definition:Multiplication/Complex Numbers",
"Definition:Field of Complex Numbers",
"Complex Numbers under Addition f... |
proofwiki-17320 | Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space | Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Let $D \subseteq X$ be a subset of $X$.
Let $D^-$ be the closure of $D$.
Then $D$ is dense {{iff}} $D^- = X$. | === Necessary Condition ===
{{:Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Necessary Condition}}{{qed|lemma}} | Let $\struct {X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $D \subseteq X$ be a [[Definition:Subset|subset]] of $X$.
Let $D^-$ be the [[Definition:Closure in Normed Vector Space|closure]] of $D$.
Then $D$ is [[Definition:Everywhere Dense in Normed Vector Space|dense]] {{... | === [[Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Necessary Condition|Necessary Condition]] ===
{{:Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Necessary Condition}}{{qed|lemma}} | Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space | https://proofwiki.org/wiki/Subset_of_Normed_Vector_Space_is_Everywhere_Dense_iff_Closure_is_Normed_Vector_Space | https://proofwiki.org/wiki/Subset_of_Normed_Vector_Space_is_Everywhere_Dense_iff_Closure_is_Normed_Vector_Space | [
"Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space",
"Normed Vector Spaces",
"Denseness",
"Set Closures"
] | [
"Definition:Normed Vector Space",
"Definition:Subset",
"Definition:Closure/Normed Vector Space",
"Definition:Everywhere Dense/Normed Vector Space"
] | [
"Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Necessary Condition"
] |
proofwiki-17321 | Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Necessary Condition | Let $\struct {X, \norm {\, \cdot \,} }$ is a normed vector space.
Let $D \subseteq X$ be a subset of $X$.
Let $D^-$ be the closure of $D$.
Let $D$ be dense.
Then:
:$D^- = X$ | Let $x \in X \setminus D$.
Suppose $D$ is dense in $X$.
Then:
:$\forall n \in N : \exists d_n \in D : d_n \in \map {B_{\frac 1 n}} x$
where $\ds \map {B_{\frac 1 n}} x$ is an open ball.
Let $\sequence {d_n}_{n \mathop \in \N}$ be a sequence in $D$.
Then:
:$\forall n \in \N : \norm {x - d_n} < \frac 1 n$
Hence, $x$ is a... | Let $\struct {X, \norm {\, \cdot \,} }$ is a [[Definition:Normed Vector Space|normed vector space]].
Let $D \subseteq X$ be a [[Definition:Subset|subset]] of $X$.
Let $D^-$ be the [[Definition:Closure in Normed Vector Space|closure]] of $D$.
Let $D$ be [[Definition:Everywhere Dense in Normed Vector Space|dense]].
T... | Let $x \in X \setminus D$.
Suppose $D$ is [[Definition:Everywhere Dense in Normed Vector Space|dense]] in $X$.
Then:
:$\forall n \in N : \exists d_n \in D : d_n \in \map {B_{\frac 1 n}} x$
where $\ds \map {B_{\frac 1 n}} x$ is an [[Definition:Open Ball in Normed Vector Space|open ball]].
Let $\sequence {d_n}_{n \m... | Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Necessary Condition | https://proofwiki.org/wiki/Subset_of_Normed_Vector_Space_is_Everywhere_Dense_iff_Closure_is_Normed_Vector_Space/Necessary_Condition | https://proofwiki.org/wiki/Subset_of_Normed_Vector_Space_is_Everywhere_Dense_iff_Closure_is_Normed_Vector_Space/Necessary_Condition | [
"Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space"
] | [
"Definition:Normed Vector Space",
"Definition:Subset",
"Definition:Closure/Normed Vector Space",
"Definition:Everywhere Dense/Normed Vector Space"
] | [
"Definition:Everywhere Dense/Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Sequence",
"Definition:Limit Point/Normed Vector Space",
"Definition:Closure/Normed Vector Space"
] |
proofwiki-17322 | Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Sufficient Condition | Let $\struct {X, \norm {\, \cdot \,} }$ is a normed vector space.
Let $D \subseteq X$ be a subset of $X$.
Let $D^-$ be the closure of $D$.
Let $D^- = X$.
Then $D$ is dense. | Let $X = D^-$.
We have to show, that for every $x \in X$ there is an open ball with an element from $D^-$.
We have that $X = D \cup \paren {X \setminus D}$.
Suppose $x \in X \setminus D$.
Then $x \in D^- \setminus D$.
Hence, $x$ is a limit point of $D$.
Therefore, there is a sequence $\sequence {d_n}_{n \mathop \in \N}... | Let $\struct {X, \norm {\, \cdot \,} }$ is a [[Definition:Normed Vector Space|normed vector space]].
Let $D \subseteq X$ be a [[Definition:Subset|subset]] of $X$.
Let $D^-$ be the [[Definition:Closure in Normed Vector Space|closure]] of $D$.
Let $D^- = X$.
Then $D$ is [[Definition:Everywhere Dense in Normed Vector... | Let $X = D^-$.
We have to show, that for every $x \in X$ there is an [[Definition:Open Ball in Normed Vector Space|open ball]] with an [[Definition:Element|element]] from $D^-$.
We have that $X = D \cup \paren {X \setminus D}$.
Suppose $x \in X \setminus D$.
Then $x \in D^- \setminus D$.
Hence, $x$ is a [[Definiti... | Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Sufficient Condition | https://proofwiki.org/wiki/Subset_of_Normed_Vector_Space_is_Everywhere_Dense_iff_Closure_is_Normed_Vector_Space/Sufficient_Condition | https://proofwiki.org/wiki/Subset_of_Normed_Vector_Space_is_Everywhere_Dense_iff_Closure_is_Normed_Vector_Space/Sufficient_Condition | [
"Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space"
] | [
"Definition:Normed Vector Space",
"Definition:Subset",
"Definition:Closure/Normed Vector Space",
"Definition:Everywhere Dense/Normed Vector Space"
] | [
"Definition:Open Ball/Normed Vector Space",
"Definition:Element",
"Definition:Limit Point/Normed Vector Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Everywhere Dense/Normed Vector Space"
] |
proofwiki-17323 | Modulus of Limit/Normed Vector Space | Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $\sequence {x_n}$ be a convergent sequence in $R$ to the limit $x$.
That is, let $\ds \lim_{n \mathop \to \infty} x_n = x$.
Then:
:$\ds \lim_{n \mathop \to \infty} \norm {x_n} = \norm x$ | By the Reverse Triangle Inequality:
:$\cmod {\norm {x_n} - \norm x} \le \norm {x_n - x}$
Hence by the Squeeze Theorem and Convergent Sequence Minus Limit:
:$\norm {x_n} \to \norm x$
as $n \to \infty$.
{{Qed}} | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\sequence {x_n}$ be a [[Definition:Convergent Sequence in Normed Vector Space|convergent sequence]] in $R$ to the [[Definition:Limit of Sequence in Normed Vector Space|limit]] $x$.
That is, let $\ds \lim_{n \math... | By the [[Reverse Triangle Inequality]]:
:$\cmod {\norm {x_n} - \norm x} \le \norm {x_n - x}$
Hence by the [[Squeeze Theorem]] and [[Convergent Sequence Minus Limit]]:
:$\norm {x_n} \to \norm x$
as $n \to \infty$.
{{Qed}} | Modulus of Limit/Normed Vector Space | https://proofwiki.org/wiki/Modulus_of_Limit/Normed_Vector_Space | https://proofwiki.org/wiki/Modulus_of_Limit/Normed_Vector_Space | [
"Modulus of Limit",
"Convergent Sequences (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Limit of Sequence/Normed Vector Space"
] | [
"Reverse Triangle Inequality",
"Squeeze Theorem",
"Convergent Sequence Minus Limit"
] |
proofwiki-17324 | Matrix Entrywise Addition forms Abelian Group | Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.
Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $\struct {R, +, \circ}$.
Then $\struct {\map {\MM_R} {m, n}, +}$, where $+$ is matrix entrywise addition, is a group. | We have by definition that matrix entrywise addition is a specific instance of a Hadamard product.
By definition of a ring, the structure $\struct {R, +}$ is a group.
As $\struct {R, +}$ is a fortiori a monoid, it follows from Matrix Space Semigroup under Hadamard Product that $\struct {\map {\MM_R} {m, n}, +}$ is also... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]] whose [[Definition:Ring Zero|zero]] is $0_R$.
Let $\map {\MM_R} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $\struct {R, +, \circ}$.
Then $\struct {\map {\MM_R} {m, n}, +}$, where $+$ is [[Definition:Matrix Entryw... | We have by definition that [[Definition:Matrix Entrywise Addition|matrix entrywise addition]] is a specific instance of a [[Definition:Hadamard Product|Hadamard product]].
By definition of a [[Definition:Ring (Abstract Algebra)|ring]], the [[Definition:Algebraic Structure with One Operation|structure]] $\struct {R, +}... | Matrix Entrywise Addition forms Abelian Group | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_forms_Abelian_Group | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_forms_Abelian_Group | [
"Matrix Entrywise Addition",
"Examples of Groups",
"Matrix Entrywise Addition forms Abelian Group"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Zero",
"Definition:Matrix Space",
"Definition:Matrix Entrywise Addition",
"Definition:Group"
] | [
"Definition:Matrix Entrywise Addition",
"Definition:Hadamard Product",
"Definition:Ring (Abstract Algebra)",
"Definition:Algebraic Structure/One Operation",
"Definition:Group",
"Definition:A Fortiori",
"Definition:Monoid",
"Matrix Space Semigroup under Hadamard Product",
"Definition:Monoid",
"Defi... |
proofwiki-17325 | Negative Matrix is Inverse for Matrix Entrywise Addition over Ring | Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.
Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $\struct {R, +, \circ}$.
Let $\mathbf A$ be an element of $\map {\MM_R} {m, n}$.
Let $-\mathbf A$ be the negative of $\mathbf A$.
Then $-\mathbf A$ is the inverse for the operation $+$, where $+$ is ... | Let $\mathbf A = \sqbrk a_{m n} \in \map {\MM_R} {m, n}$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \paren {-\mathbf A}
| r = \sqbrk a_{m n} + \paren {-\sqbrk a_{m n} }
| c = Definition of $\mathbf A$
}}
{{eqn | r = \sqbrk a_{m n} + \sqbrk {-a}_{m n}
| c = {{Defof|Negative Matrix over Ring}}
}}
{{eq... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]] whose [[Definition:Ring Zero|zero]] is $0_R$.
Let $\map {\MM_R} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $\struct {R, +, \circ}$.
Let $\mathbf A$ be an [[Definition:Element|element]] of $\map {\MM_R} {m, n}$.
... | Let $\mathbf A = \sqbrk a_{m n} \in \map {\MM_R} {m, n}$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \paren {-\mathbf A}
| r = \sqbrk a_{m n} + \paren {-\sqbrk a_{m n} }
| c = Definition of $\mathbf A$
}}
{{eqn | r = \sqbrk a_{m n} + \sqbrk {-a}_{m n}
| c = {{Defof|Negative Matrix over Ring}}
}}
{{... | Negative Matrix is Inverse for Matrix Entrywise Addition over Ring | https://proofwiki.org/wiki/Negative_Matrix_is_Inverse_for_Matrix_Entrywise_Addition_over_Ring | https://proofwiki.org/wiki/Negative_Matrix_is_Inverse_for_Matrix_Entrywise_Addition_over_Ring | [
"Negative Matrices"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Zero",
"Definition:Matrix Space",
"Definition:Element",
"Definition:Negative Matrix",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Matrix Entrywise Addition/Ring"
] | [
"Zero Matrix is Identity for Matrix Entrywise Addition over Ring"
] |
proofwiki-17326 | Zero Matrix is Identity for Matrix Entrywise Addition over Ring | Let $\struct {R, +, \circ}$ be a ring.
Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$.
Let $\mathbf 0_R = \sqbrk {0_R}_{m n}$ be the zero matrix of $\map {\MM_R} {m, n}$.
Then $\mathbf 0_R$ is the identity element for matrix entrywise addition. | Let $\mathbf A = \sqbrk a_{m n} \in \map {\MM_R} {m, n}$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \mathbf 0_R
| r = \sqbrk a_{m n} + \sqbrk {0_R}_{m n}
| c = Definition of $\mathbf A$ and $\mathbf 0_R$
}}
{{eqn | r = \sqbrk {a + 0_R}_{m n}
| c = {{Defof|Matrix Entrywise Addition}}
}}
{{eqn | r = \... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\map {\MM_R} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $R$.
Let $\mathbf 0_R = \sqbrk {0_R}_{m n}$ be the [[Definition:Zero Matrix over Ring|zero matrix]] of $\map {\MM_R} {m, n}$.
Then $\mathbf 0_R$ is ... | Let $\mathbf A = \sqbrk a_{m n} \in \map {\MM_R} {m, n}$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \mathbf 0_R
| r = \sqbrk a_{m n} + \sqbrk {0_R}_{m n}
| c = Definition of $\mathbf A$ and $\mathbf 0_R$
}}
{{eqn | r = \sqbrk {a + 0_R}_{m n}
| c = {{Defof|Matrix Entrywise Addition}}
}}
{{eqn | r =... | Zero Matrix is Identity for Matrix Entrywise Addition over Ring/Proof 1 | https://proofwiki.org/wiki/Zero_Matrix_is_Identity_for_Matrix_Entrywise_Addition_over_Ring | https://proofwiki.org/wiki/Zero_Matrix_is_Identity_for_Matrix_Entrywise_Addition_over_Ring/Proof_1 | [
"Matrix Entrywise Addition",
"Zero Matrix",
"Zero Matrix is Identity for Matrix Entrywise Addition"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Matrix Space",
"Definition:Zero Matrix/Ring",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Matrix Entrywise Addition/Ring"
] | [] |
proofwiki-17327 | Zero Matrix is Identity for Matrix Entrywise Addition over Ring | Let $\struct {R, +, \circ}$ be a ring.
Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$.
Let $\mathbf 0_R = \sqbrk {0_R}_{m n}$ be the zero matrix of $\map {\MM_R} {m, n}$.
Then $\mathbf 0_R$ is the identity element for matrix entrywise addition. | By definition, matrix entrywise addition is the '''Hadamard product''' with respect to ring addition.
We have from {{Ring-axiom|A3}} that the identity element of ring addition is the ring zero $0_R$.
The result then follows directly from Zero Matrix is Identity for Hadamard Product.
{{qed}} | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\map {\MM_R} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $R$.
Let $\mathbf 0_R = \sqbrk {0_R}_{m n}$ be the [[Definition:Zero Matrix over Ring|zero matrix]] of $\map {\MM_R} {m, n}$.
Then $\mathbf 0_R$ is ... | By definition, [[Definition:Matrix Entrywise Addition|matrix entrywise addition]] is the '''[[Definition:Hadamard Product|Hadamard product]]''' with respect to [[Definition:Ring Addition|ring addition]].
We have from {{Ring-axiom|A3}} that the [[Definition:Identity Element|identity element]] of [[Definition:Ring Addit... | Zero Matrix is Identity for Matrix Entrywise Addition over Ring/Proof 2 | https://proofwiki.org/wiki/Zero_Matrix_is_Identity_for_Matrix_Entrywise_Addition_over_Ring | https://proofwiki.org/wiki/Zero_Matrix_is_Identity_for_Matrix_Entrywise_Addition_over_Ring/Proof_2 | [
"Matrix Entrywise Addition",
"Zero Matrix",
"Zero Matrix is Identity for Matrix Entrywise Addition"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Matrix Space",
"Definition:Zero Matrix/Ring",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Matrix Entrywise Addition/Ring"
] | [
"Definition:Matrix Entrywise Addition",
"Definition:Hadamard Product",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Ring Zero",
"Zero Matrix is Identity for Hadamard Product"
] |
proofwiki-17328 | Properties of Matrix Entrywise Addition over Ring | Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.
Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $S$ over an algebraic structure $\struct {R, +, \circ}$.
Let $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$.
Let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\ma... | === Matrix Entrywise Addition over Ring is Closed ===
{{:Matrix Entrywise Addition over Ring is Closed}} | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]] whose [[Definition:Ring Zero|zero]] is $0_R$.
Let $\map {\MM_R} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $S$ over an [[Definition:Algebraic Structure with Two Operations|algebraic structure]] $\struct {R, +, \ci... | === [[Matrix Entrywise Addition over Ring is Closed]] ===
{{:Matrix Entrywise Addition over Ring is Closed}} | Properties of Matrix Entrywise Addition over Ring | https://proofwiki.org/wiki/Properties_of_Matrix_Entrywise_Addition_over_Ring | https://proofwiki.org/wiki/Properties_of_Matrix_Entrywise_Addition_over_Ring | [
"Matrix Entrywise Addition"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Zero",
"Definition:Matrix Space",
"Definition:Algebraic Structure/Two Operations",
"Definition:Matrix Entrywise Addition",
"Definition:Matrix Entrywise Addition/Ring",
"Definition:Closure (Abstract Algebra)",
"Definition:Associative Operation",
... | [
"Matrix Entrywise Addition over Ring is Closed"
] |
proofwiki-17329 | Matrix Entrywise Addition is Associative | Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.
For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
The operation $+$ is associative on $\map \MM {m, n}$.
That is:
:$\paren {\math... | From:
:Integers form Ring
:Rational Numbers form Ring
:Real Numbers form Ring
:Complex Numbers form Ring
the standard number systems $\Z$, $\Q$, $\R$ and $\C$ are rings.
Hence we can apply Matrix Entrywise Addition over Ring is Associative.
{{qed|lemma}}
The above cannot be applied to the natural numbers $\N$, as they ... | Let $\map \MM {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over one of the [[Definition:Standard Number System|standard number systems]].
For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the [[Definition:Matrix Entrywise Addition|matrix entrywise sum]] of ... | From:
:[[Integers form Ring]]
:[[Rational Numbers form Ring]]
:[[Real Numbers form Ring]]
:[[Complex Numbers form Ring]]
the [[Definition:Standard Number System|standard number systems]] $\Z$, $\Q$, $\R$ and $\C$ are [[Definition:Ring (Abstract Algebra)|rings]].
Hence we can apply [[Matrix Entrywise Addition over Rin... | Matrix Entrywise Addition is Associative/Proof 1 | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_is_Associative | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_is_Associative/Proof_1 | [
"Matrix Entrywise Addition is Associative",
"Matrix Entrywise Addition",
"Examples of Associative Operations"
] | [
"Definition:Matrix Space",
"Definition:Number",
"Definition:Matrix Entrywise Addition",
"Definition:Associative Operation"
] | [
"Integers form Commutative Ring",
"Rational Numbers form Ring",
"Real Numbers form Ring",
"Complex Numbers form Ring",
"Definition:Number",
"Definition:Ring (Abstract Algebra)",
"Matrix Entrywise Addition over Ring is Associative",
"Definition:Natural Numbers",
"Definition:Ring (Abstract Algebra)",
... |
proofwiki-17330 | Matrix Entrywise Addition is Associative | Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.
For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
The operation $+$ is associative on $\map \MM {m, n}$.
That is:
:$\paren {\math... | Let $\mathbf A = \sqbrk a_{m n}$, $\mathbf B = \sqbrk b_{m n}$ and $\mathbf C = \sqbrk c_{m n}$ be matrices whose order is $m \times n$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {\mathbf A + \mathbf B} + \mathbf C
| r = \paren {\sqbrk a_{m n} + \sqbrk b_{m n} } + \sqbrk c_{m n}
| c = Definition of $\mathbf A$... | Let $\map \MM {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over one of the [[Definition:Standard Number System|standard number systems]].
For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the [[Definition:Matrix Entrywise Addition|matrix entrywise sum]] of ... | Let $\mathbf A = \sqbrk a_{m n}$, $\mathbf B = \sqbrk b_{m n}$ and $\mathbf C = \sqbrk c_{m n}$ be [[Definition:Matrix|matrices]] whose [[Definition:Order of Matrix|order]] is $m \times n$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {\mathbf A + \mathbf B} + \mathbf C
| r = \paren {\sqbrk a_{m n} + \sqbrk b_{m n} }... | Matrix Entrywise Addition is Associative/Proof 2 | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_is_Associative | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_is_Associative/Proof_2 | [
"Matrix Entrywise Addition is Associative",
"Matrix Entrywise Addition",
"Examples of Associative Operations"
] | [
"Definition:Matrix Space",
"Definition:Number",
"Definition:Matrix Entrywise Addition",
"Definition:Associative Operation"
] | [
"Definition:Matrix",
"Definition:Matrix/Order",
"Associative Law of Addition"
] |
proofwiki-17331 | Matrix Entrywise Addition over Ring is Commutative | Let $\struct {R, +, \circ}$ be a ring.
Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$.
For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
The operation $+$ is commutative on $\map {\MM_R} {m, n}$.
That... | Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be elements of the $m \times n$ matrix space over $R$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \mathbf B
| r = \sqbrk a_{m n} + \sqbrk b_{m n}
| c = Definition of $\mathbf A$ and $\mathbf B$
}}
{{eqn | r = \sqbrk {a + b}_{m n}
| c =... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\map {\MM_R} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $R$.
For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the [[Definition:Matrix Entrywise Addition over Rin... | Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be [[Definition:Element|elements]] of the [[Definition:Matrix Space|$m \times n$ matrix space]] over $R$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \mathbf B
| r = \sqbrk a_{m n} + \sqbrk b_{m n}
| c = Definition of $\mathbf A$ and $\math... | Matrix Entrywise Addition over Ring is Commutative/Proof 1 | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_over_Ring_is_Commutative | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_over_Ring_is_Commutative/Proof_1 | [
"Matrix Entrywise Addition",
"Examples of Commutative Operations",
"Matrix Entrywise Addition is Commutative"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Matrix Space",
"Definition:Matrix Entrywise Addition/Ring",
"Definition:Commutative/Operation"
] | [
"Definition:Element",
"Definition:Matrix Space"
] |
proofwiki-17332 | Matrix Entrywise Addition over Ring is Commutative | Let $\struct {R, +, \circ}$ be a ring.
Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$.
For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
The operation $+$ is commutative on $\map {\MM_R} {m, n}$.
That... | By definition, matrix entrywise addition is the '''Hadamard product''' of $\mathbf A$ and $\mathbf B$ with respect to ring addition.
We have from {{Ring-axiom|A2}} that ring addition is commutative.
The result then follows directly from Commutativity of Hadamard Product.
{{qed}} | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\map {\MM_R} {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $R$.
For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the [[Definition:Matrix Entrywise Addition over Rin... | By definition, [[Definition:Matrix Entrywise Addition|matrix entrywise addition]] is the '''[[Definition:Hadamard Product|Hadamard product]]''' of $\mathbf A$ and $\mathbf B$ with respect to [[Definition:Ring Addition|ring addition]].
We have from {{Ring-axiom|A2}} that [[Definition:Ring Addition|ring addition]] is [[... | Matrix Entrywise Addition over Ring is Commutative/Proof 2 | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_over_Ring_is_Commutative | https://proofwiki.org/wiki/Matrix_Entrywise_Addition_over_Ring_is_Commutative/Proof_2 | [
"Matrix Entrywise Addition",
"Examples of Commutative Operations",
"Matrix Entrywise Addition is Commutative"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Matrix Space",
"Definition:Matrix Entrywise Addition/Ring",
"Definition:Commutative/Operation"
] | [
"Definition:Matrix Entrywise Addition",
"Definition:Hadamard Product",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Commutative/Operation",
"Commutativity of Hadamard Product"
] |
proofwiki-17333 | Zero Matrix is Identity for Matrix Entrywise Addition | Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.
Let $\mathbf 0 = \sqbrk 0_{m n}$ be the zero matrix of $\map \MM {m, n}$.
Then $\mathbf 0$ is the identity element for matrix entrywise addition. | Let $\mathbf A = \sqbrk a_{m n} \in \map {\MM_R} {m, n}$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \mathbf 0_R
| r = \sqbrk a_{m n} + \sqbrk {0_R}_{m n}
| c = Definition of $\mathbf A$ and $\mathbf 0_R$
}}
{{eqn | r = \sqbrk {a + 0_R}_{m n}
| c = {{Defof|Matrix Entrywise Addition}}
}}
{{eqn | r = \... | Let $\map \MM {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over one of the [[Definition:Standard Number System|standard number systems]].
Let $\mathbf 0 = \sqbrk 0_{m n}$ be the [[Definition:Zero Matrix|zero matrix]] of $\map \MM {m, n}$.
Then $\mathbf 0$ is the [[Definition:Identity Element|id... | Let $\mathbf A = \sqbrk a_{m n} \in \map {\MM_R} {m, n}$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \mathbf 0_R
| r = \sqbrk a_{m n} + \sqbrk {0_R}_{m n}
| c = Definition of $\mathbf A$ and $\mathbf 0_R$
}}
{{eqn | r = \sqbrk {a + 0_R}_{m n}
| c = {{Defof|Matrix Entrywise Addition}}
}}
{{eqn | r =... | Zero Matrix is Identity for Matrix Entrywise Addition over Ring/Proof 1 | https://proofwiki.org/wiki/Zero_Matrix_is_Identity_for_Matrix_Entrywise_Addition | https://proofwiki.org/wiki/Zero_Matrix_is_Identity_for_Matrix_Entrywise_Addition_over_Ring/Proof_1 | [
"Matrix Entrywise Addition",
"Zero Matrix",
"Zero Matrix is Identity for Matrix Entrywise Addition"
] | [
"Definition:Matrix Space",
"Definition:Number",
"Definition:Zero Matrix",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Matrix Entrywise Addition"
] | [] |
proofwiki-17334 | Zero Matrix is Identity for Matrix Entrywise Addition | Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.
Let $\mathbf 0 = \sqbrk 0_{m n}$ be the zero matrix of $\map \MM {m, n}$.
Then $\mathbf 0$ is the identity element for matrix entrywise addition. | By definition, matrix entrywise addition is the '''Hadamard product''' with respect to ring addition.
We have from {{Ring-axiom|A3}} that the identity element of ring addition is the ring zero $0_R$.
The result then follows directly from Zero Matrix is Identity for Hadamard Product.
{{qed}} | Let $\map \MM {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over one of the [[Definition:Standard Number System|standard number systems]].
Let $\mathbf 0 = \sqbrk 0_{m n}$ be the [[Definition:Zero Matrix|zero matrix]] of $\map \MM {m, n}$.
Then $\mathbf 0$ is the [[Definition:Identity Element|id... | By definition, [[Definition:Matrix Entrywise Addition|matrix entrywise addition]] is the '''[[Definition:Hadamard Product|Hadamard product]]''' with respect to [[Definition:Ring Addition|ring addition]].
We have from {{Ring-axiom|A3}} that the [[Definition:Identity Element|identity element]] of [[Definition:Ring Addit... | Zero Matrix is Identity for Matrix Entrywise Addition over Ring/Proof 2 | https://proofwiki.org/wiki/Zero_Matrix_is_Identity_for_Matrix_Entrywise_Addition | https://proofwiki.org/wiki/Zero_Matrix_is_Identity_for_Matrix_Entrywise_Addition_over_Ring/Proof_2 | [
"Matrix Entrywise Addition",
"Zero Matrix",
"Zero Matrix is Identity for Matrix Entrywise Addition"
] | [
"Definition:Matrix Space",
"Definition:Number",
"Definition:Zero Matrix",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Matrix Entrywise Addition"
] | [
"Definition:Matrix Entrywise Addition",
"Definition:Hadamard Product",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Ring Zero",
"Zero Matrix is Identity for Hadamard Product"
] |
proofwiki-17335 | Zero Matrix is Identity for Matrix Entrywise Addition | Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.
Let $\mathbf 0 = \sqbrk 0_{m n}$ be the zero matrix of $\map \MM {m, n}$.
Then $\mathbf 0$ is the identity element for matrix entrywise addition. | From:
:Integers form Ring
:Rational Numbers form Ring
:Real Numbers form Ring
:Complex Numbers form Ring
the standard number systems $\Z$, $\Q$, $\R$ and $\C$ are rings whose zero is the number $0$ (zero).
Hence we can apply Zero Matrix is Identity for Matrix Entrywise Addition over Ring.
{{qed|lemma}}
The above cannot... | Let $\map \MM {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over one of the [[Definition:Standard Number System|standard number systems]].
Let $\mathbf 0 = \sqbrk 0_{m n}$ be the [[Definition:Zero Matrix|zero matrix]] of $\map \MM {m, n}$.
Then $\mathbf 0$ is the [[Definition:Identity Element|id... | From:
:[[Integers form Ring]]
:[[Rational Numbers form Ring]]
:[[Real Numbers form Ring]]
:[[Complex Numbers form Ring]]
the [[Definition:Standard Number System|standard number systems]] $\Z$, $\Q$, $\R$ and $\C$ are [[Definition:Ring (Abstract Algebra)|rings]] whose [[Definition:Zero Element|zero]] is the [[Definitio... | Zero Matrix is Identity for Matrix Entrywise Addition/Proof 1 | https://proofwiki.org/wiki/Zero_Matrix_is_Identity_for_Matrix_Entrywise_Addition | https://proofwiki.org/wiki/Zero_Matrix_is_Identity_for_Matrix_Entrywise_Addition/Proof_1 | [
"Matrix Entrywise Addition",
"Zero Matrix",
"Zero Matrix is Identity for Matrix Entrywise Addition"
] | [
"Definition:Matrix Space",
"Definition:Number",
"Definition:Zero Matrix",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Matrix Entrywise Addition"
] | [
"Integers form Commutative Ring",
"Rational Numbers form Ring",
"Real Numbers form Ring",
"Complex Numbers form Ring",
"Definition:Number",
"Definition:Ring (Abstract Algebra)",
"Definition:Zero Element",
"Definition:Zero (Number)",
"Zero Matrix is Identity for Matrix Entrywise Addition over Ring",
... |
proofwiki-17336 | Zero Matrix is Identity for Matrix Entrywise Addition | Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.
Let $\mathbf 0 = \sqbrk 0_{m n}$ be the zero matrix of $\map \MM {m, n}$.
Then $\mathbf 0$ is the identity element for matrix entrywise addition. | Let $\mathbf A = \sqbrk a_{m n} \in \map \MM {m, n}$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \mathbf 0
| r = \sqbrk a_{m n} + \sqbrk 0_{m n}
| c = Definition of $\mathbf A$ and $\mathbf 0_R$
}}
{{eqn | r = \sqbrk {a + 0}_{m n}
| c = {{Defof|Matrix Entrywise Addition}}
}}
{{eqn | r = \sqbrk a_{m n... | Let $\map \MM {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over one of the [[Definition:Standard Number System|standard number systems]].
Let $\mathbf 0 = \sqbrk 0_{m n}$ be the [[Definition:Zero Matrix|zero matrix]] of $\map \MM {m, n}$.
Then $\mathbf 0$ is the [[Definition:Identity Element|id... | Let $\mathbf A = \sqbrk a_{m n} \in \map \MM {m, n}$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \mathbf 0
| r = \sqbrk a_{m n} + \sqbrk 0_{m n}
| c = Definition of $\mathbf A$ and $\mathbf 0_R$
}}
{{eqn | r = \sqbrk {a + 0}_{m n}
| c = {{Defof|Matrix Entrywise Addition}}
}}
{{eqn | r = \sqbrk a_{m... | Zero Matrix is Identity for Matrix Entrywise Addition/Proof 2 | https://proofwiki.org/wiki/Zero_Matrix_is_Identity_for_Matrix_Entrywise_Addition | https://proofwiki.org/wiki/Zero_Matrix_is_Identity_for_Matrix_Entrywise_Addition/Proof_2 | [
"Matrix Entrywise Addition",
"Zero Matrix",
"Zero Matrix is Identity for Matrix Entrywise Addition"
] | [
"Definition:Matrix Space",
"Definition:Number",
"Definition:Zero Matrix",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Matrix Entrywise Addition"
] | [
"Identity Element of Addition on Numbers",
"Identity Element of Addition on Numbers"
] |
proofwiki-17337 | Matrix Scalar Product Distributes over Number Addition | Let $\GF$ denote one of the standard number systems.
Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\GF$.
For $\mathbf A \in \map \MM {m, n}$ and $\lambda \in \GF$, let $\lambda \mathbf A$ be defined as the matrix scalar product of $\lambda$ and $\mathbf A$.
The matrix scalar product is distributive on $\ma... | {{begin-eqn}}
{{eqn | l = \paren {\lambda + \mu} \mathbf A
| r = \paren {\lambda + \mu} \sqbrk a_{m n}
| c = Definition of $\mathbf A$
}}
{{eqn | r = \sqbrk {\paren {\lambda + \mu} a}_{m n}
| c = {{Defof|Matrix Scalar Product}}
}}
{{eqn | r = \sqbrk {\lambda a + \mu a}_{m n}
| c = Distributive P... | Let $\GF$ denote one of the [[Definition:Standard Number System|standard number systems]].
Let $\map \MM {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $\GF$.
For $\mathbf A \in \map \MM {m, n}$ and $\lambda \in \GF$, let $\lambda \mathbf A$ be defined as the [[Definition:Matrix Scalar Produc... | {{begin-eqn}}
{{eqn | l = \paren {\lambda + \mu} \mathbf A
| r = \paren {\lambda + \mu} \sqbrk a_{m n}
| c = Definition of $\mathbf A$
}}
{{eqn | r = \sqbrk {\paren {\lambda + \mu} a}_{m n}
| c = {{Defof|Matrix Scalar Product}}
}}
{{eqn | r = \sqbrk {\lambda a + \mu a}_{m n}
| c = [[Distributive... | Matrix Scalar Product Distributes over Number Addition/Proof | https://proofwiki.org/wiki/Matrix_Scalar_Product_Distributes_over_Number_Addition | https://proofwiki.org/wiki/Matrix_Scalar_Product_Distributes_over_Number_Addition/Proof | [
"Matrix Scalar Product Distributes over Number Addition",
"Matrix Scalar Product",
"Addition",
"Examples of Distributive Operations"
] | [
"Definition:Number",
"Definition:Matrix Space",
"Definition:Matrix Scalar Product",
"Definition:Matrix Scalar Product",
"Definition:Distributive Operation"
] | [
"Distributive Laws/Arithmetic"
] |
proofwiki-17338 | Matrix Scalar Product with Zero gives Zero Matrix | Let $\Bbb F$ denote one of the standard number systems.
Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$.
For $\mathbf A \in \map \MM {m, n}$ and $\lambda$ \in $\Bbb F$, let $\lambda \mathbf A$ be defined as the matrix scalar product of $\lambda$ and $\mathbf A$.
When $\lambda = 0$, we have for all $\... | {{begin-eqn}}
{{eqn | l = 0 \mathbf A
| r = 0 \sqbrk a_{m n}
| c = Definition of $\mathbf A$
}}
{{eqn | r = \sqbrk {0 a}_{m n}
| c = {{Defof|Matrix Scalar Product}}
}}
{{eqn | r = \sqbrk 0_{m n}
| c = Zero is Zero Element for Multiplication
}}
{{eqn | r = \mathbf 0
| c = {{Defof|Zero Matri... | Let $\Bbb F$ denote one of the [[Definition:Standard Number System|standard number systems]].
Let $\map \MM {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $\Bbb F$.
For $\mathbf A \in \map \MM {m, n}$ and $\lambda$ \in $\Bbb F$, let $\lambda \mathbf A$ be defined as the [[Definition:Matrix Sc... | {{begin-eqn}}
{{eqn | l = 0 \mathbf A
| r = 0 \sqbrk a_{m n}
| c = Definition of $\mathbf A$
}}
{{eqn | r = \sqbrk {0 a}_{m n}
| c = {{Defof|Matrix Scalar Product}}
}}
{{eqn | r = \sqbrk 0_{m n}
| c = [[Zero is Zero Element for Multiplication]]
}}
{{eqn | r = \mathbf 0
| c = {{Defof|Zero M... | Matrix Scalar Product with Zero gives Zero Matrix | https://proofwiki.org/wiki/Matrix_Scalar_Product_with_Zero_gives_Zero_Matrix | https://proofwiki.org/wiki/Matrix_Scalar_Product_with_Zero_gives_Zero_Matrix | [
"Matrix Scalar Product",
"Addition",
"Examples of Distributive Operations"
] | [
"Definition:Number",
"Definition:Matrix Space",
"Definition:Matrix Scalar Product",
"Definition:Zero Matrix"
] | [
"Zero is Zero Element for Multiplication"
] |
proofwiki-17339 | Negative Matrix is Inverse for Matrix Entrywise Addition | Let $\Bbb F$ denote one of the standard number systems.
Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$.
Let $\mathbf A$ be an element of $\map \MM {m, n}$.
Let $-\mathbf A$ be the negative of $\mathbf A$.
Then $-\mathbf A$ is the inverse for the operation $+$, where $+$ is matrix entrywise addition. | Let $\mathbf A = \sqbrk a_{m n} \in \map \MM {m, n}$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \paren {-\mathbf A}
| r = \sqbrk a_{m n} + \paren {-\sqbrk a_{m n} }
| c = Definition of $\mathbf A$
}}
{{eqn | r = \sqbrk a_{m n} + \sqbrk {-a}_{m n}
| c = {{Defof|Negative Matrix}}
}}
{{eqn | r = \sqbrk... | Let $\Bbb F$ denote one of the [[Definition:Standard Number System|standard number systems]].
Let $\map \MM {m, n}$ be a [[Definition:Matrix Space|$m \times n$ matrix space]] over $\Bbb F$.
Let $\mathbf A$ be an [[Definition:Element|element]] of $\map \MM {m, n}$.
Let $-\mathbf A$ be the [[Definition:Negative Matrix... | Let $\mathbf A = \sqbrk a_{m n} \in \map \MM {m, n}$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A + \paren {-\mathbf A}
| r = \sqbrk a_{m n} + \paren {-\sqbrk a_{m n} }
| c = Definition of $\mathbf A$
}}
{{eqn | r = \sqbrk a_{m n} + \sqbrk {-a}_{m n}
| c = {{Defof|Negative Matrix}}
}}
{{eqn | r = \sqb... | Negative Matrix is Inverse for Matrix Entrywise Addition | https://proofwiki.org/wiki/Negative_Matrix_is_Inverse_for_Matrix_Entrywise_Addition | https://proofwiki.org/wiki/Negative_Matrix_is_Inverse_for_Matrix_Entrywise_Addition | [
"Negative Matrices"
] | [
"Definition:Number",
"Definition:Matrix Space",
"Definition:Element",
"Definition:Negative Matrix",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Definition:Matrix Entrywise Addition"
] | [
"Zero Matrix is Identity for Matrix Entrywise Addition"
] |
proofwiki-17340 | Zero Matrix is Zero for Matrix Multiplication | Let $\struct {R, +, \times}$ be a ring.
Let $\mathbf A$ be a matrix over $R$ of order $m \times n$
Let $\mathbf 0$ be a zero matrix whose order is such that either:
:$\mathbf {0 A}$ is defined
or:
:$\mathbf {A 0}$ is defined
or both.
Then:
:$\mathbf {0 A} = \mathbf 0$
or:
:$\mathbf {A 0} = \mathbf 0$
whenever they are ... | Let $\mathbf A = \sqbrk a_{m n}$ be matrices.
Let $\mathbf {0 A}$ be defined.
Then $\mathbf 0$ is of order $r \times m$ for $r \in \Z_{>0}$.
Thus we have:
{{begin-eqn}}
{{eqn | l = \mathbf {0 A}
| r = \mathbf C
| c =
}}
{{eqn | l = \sqbrk 0_{r m} \sqbrk a_{m n}
| r = \sqbrk c_{r n}
| c = D... | Let $\struct {R, +, \times}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\mathbf A$ be a [[Definition:Matrix|matrix]] over $R$ of [[Definition:Order of Matrix|order]] $m \times n$
Let $\mathbf 0$ be a [[Definition:Zero Matrix|zero matrix]] whose [[Definition:Order of Matrix|order]] is such that either:
:$\... | Let $\mathbf A = \sqbrk a_{m n}$ be [[Definition:Matrix|matrices]].
Let $\mathbf {0 A}$ be defined.
Then $\mathbf 0$ is of [[Definition:Order of Matrix|order]] $r \times m$ for $r \in \Z_{>0}$.
Thus we have:
{{begin-eqn}}
{{eqn | l = \mathbf {0 A}
| r = \mathbf C
| c =
}}
{{eqn | l = \sqbrk 0_{r ... | Zero Matrix is Zero for Matrix Multiplication | https://proofwiki.org/wiki/Zero_Matrix_is_Zero_for_Matrix_Multiplication | https://proofwiki.org/wiki/Zero_Matrix_is_Zero_for_Matrix_Multiplication | [
"Conventional Matrix Multiplication",
"Zero Matrix"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Matrix",
"Definition:Matrix/Order",
"Definition:Zero Matrix",
"Definition:Matrix/Order",
"Definition:Matrix/Order",
"Definition:Matrix/Order",
"Definition:Matrix"
] | [
"Definition:Matrix",
"Definition:Matrix/Order",
"Definition:Zero Matrix",
"Definition:Matrix/Order",
"Definition:Matrix/Order",
"Definition:Zero Matrix",
"Definition:Matrix/Order",
"Definition:Matrix/Order"
] |
proofwiki-17341 | Unit Matrix is Identity for Matrix Multiplication | Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\map {\MM_R} n$ denote the metric space of square matrices of order $n$ over $R$.
Let $\mathbf I_n$ denote the unit matrix of order $n$:
Then:
:$\forall \mathbf A \in \map {\MM_R} n: \ma... | === Lemma: Left Identity ===
{{:Unit Matrix is Identity for Matrix Multiplication/Left}}{{qed|lemma}} | Let $R$ be a [[Definition:Ring with Unity|ring with unity]] whose [[Definition:Ring Zero|zero]] is $0_R$ and whose [[Definition:Unity of Ring|unity]] is $1_R$.
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\map {\MM_R} n$ denote the [[Definition:Matrix Space|metri... | === [[Unit Matrix is Identity for Matrix Multiplication/Left|Lemma: Left Identity]] ===
{{:Unit Matrix is Identity for Matrix Multiplication/Left}}{{qed|lemma}} | Unit Matrix is Identity for Matrix Multiplication | https://proofwiki.org/wiki/Unit_Matrix_is_Identity_for_Matrix_Multiplication | https://proofwiki.org/wiki/Unit_Matrix_is_Identity_for_Matrix_Multiplication | [
"Conventional Matrix Multiplication",
"Unit Matrices",
"Unit Matrix is Identity for Matrix Multiplication"
] | [
"Definition:Ring with Unity",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Strictly Positive/Integer",
"Definition:Matrix Space",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Unit Matrix",
"Definition:Matrix/Square Matrix/... | [
"Unit Matrix is Identity for Matrix Multiplication/Left"
] |
proofwiki-17342 | Unit Matrix is Identity for Matrix Multiplication/Left | Let $\map {\MM_R} {m, n}$ denote the $m \times n$ metric space over $R$.
Let $I_m$ denote the unit matrix of order $m$.
Then:
:$\forall \mathbf A \in \map {\MM_R} {m, n}: \mathbf I_m \mathbf A = \mathbf A$ | Let $\sqbrk a_{m n} \in \map {\MM_R} {m, n}$.
Let $\sqbrk b_{m n} = \mathbf I_m \sqbrk a_{m n}$.
Then:
{{begin-eqn}}
{{eqn | q = \forall i \in \closedint 1 m, j \in \closedint 1 n
| l = b_{i j}
| r = \sum_{k \mathop = 1}^m \delta_{i k} a_{k j}
| c = where $\delta_{i k}$ is the Kronecker delta: $\delta... | Let $\map {\MM_R} {m, n}$ denote the [[Definition:Matrix Space|$m \times n$ metric space]] over $R$.
Let $I_m$ denote the [[Definition:Unit Matrix|unit matrix]] of [[Definition:Order of Square Matrix|order]] $m$.
Then:
:$\forall \mathbf A \in \map {\MM_R} {m, n}: \mathbf I_m \mathbf A = \mathbf A$ | Let $\sqbrk a_{m n} \in \map {\MM_R} {m, n}$.
Let $\sqbrk b_{m n} = \mathbf I_m \sqbrk a_{m n}$.
Then:
{{begin-eqn}}
{{eqn | q = \forall i \in \closedint 1 m, j \in \closedint 1 n
| l = b_{i j}
| r = \sum_{k \mathop = 1}^m \delta_{i k} a_{k j}
| c = where $\delta_{i k}$ is the [[Definition:Kronecke... | Unit Matrix is Identity for Matrix Multiplication/Left | https://proofwiki.org/wiki/Unit_Matrix_is_Identity_for_Matrix_Multiplication/Left | https://proofwiki.org/wiki/Unit_Matrix_is_Identity_for_Matrix_Multiplication/Left | [
"Unit Matrix is Identity for Matrix Multiplication"
] | [
"Definition:Matrix Space",
"Definition:Unit Matrix",
"Definition:Matrix/Square Matrix/Order"
] | [
"Definition:Kronecker Delta",
"Definition:Identity (Abstract Algebra)/Left Identity",
"Category:Unit Matrix is Identity for Matrix Multiplication"
] |
proofwiki-17343 | Unit Matrix is Identity for Matrix Multiplication/Right | Let $\map {\MM_R} {m, n}$ denote the $m \times n$ metric space over $R$.
Let $I_n$ denote the unit matrix of order $n$.
Then:
:$\forall \mathbf A \in \map {\MM_R} {m, n}: \mathbf A \mathbf I_n = \mathbf A$ | Let $\sqbrk a_{m n} \in \map {\MM_R} {m, n}$.
Let $\sqbrk b_{m n} = \sqbrk a_{m n} \mathbf I_n$.
Then:
{{begin-eqn}}
{{eqn | q = \forall i \in \closedint 1 m, j \in \closedint 1 n
| l = b_{i j}
| r = \sum_{k \mathop = 1}^n a_{i k} \delta_{k j}
| c = where $\delta_{k j}$ is the Kronecker delta: $\delta... | Let $\map {\MM_R} {m, n}$ denote the [[Definition:Matrix Space|$m \times n$ metric space]] over $R$.
Let $I_n$ denote the [[Definition:Unit Matrix|unit matrix]] of [[Definition:Order of Square Matrix|order]] $n$.
Then:
:$\forall \mathbf A \in \map {\MM_R} {m, n}: \mathbf A \mathbf I_n = \mathbf A$ | Let $\sqbrk a_{m n} \in \map {\MM_R} {m, n}$.
Let $\sqbrk b_{m n} = \sqbrk a_{m n} \mathbf I_n$.
Then:
{{begin-eqn}}
{{eqn | q = \forall i \in \closedint 1 m, j \in \closedint 1 n
| l = b_{i j}
| r = \sum_{k \mathop = 1}^n a_{i k} \delta_{k j}
| c = where $\delta_{k j}$ is the [[Definition:Kronecke... | Unit Matrix is Identity for Matrix Multiplication/Right | https://proofwiki.org/wiki/Unit_Matrix_is_Identity_for_Matrix_Multiplication/Right | https://proofwiki.org/wiki/Unit_Matrix_is_Identity_for_Matrix_Multiplication/Right | [
"Unit Matrix is Identity for Matrix Multiplication"
] | [
"Definition:Matrix Space",
"Definition:Unit Matrix",
"Definition:Matrix/Square Matrix/Order"
] | [
"Definition:Kronecker Delta",
"Definition:Identity (Abstract Algebra)/Right Identity",
"Category:Unit Matrix is Identity for Matrix Multiplication"
] |
proofwiki-17344 | Left and Right Inverses of Square Matrix over Field are Equal | Let $\Bbb F$ be a field, usually one of the standard number fields $\Q$, $\R$ or $\C$.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\map \MM n$ denote the matrix space of order $n$ square matrices over $\Bbb F$.
Let $\mathbf B$ be a left inverse matrix of $\mathbf A$.
Then $\mathbf B$ is also a right inve... | Consider the algebraic structure $\struct {\map \MM {m, n}, +, \circ}$, where:
:$+$ denotes matrix entrywise addition
:$\circ$ denotes (conventional) matrix multiplication.
From Ring of Square Matrices over Field is Ring with Unity, $\struct {\map \MM {m, n}, +, \circ}$ is a ring with unity.
Hence a fortiori $\struct {... | Let $\Bbb F$ be a [[Definition:Field (Abstract Algebra)|field]], usually one of the [[Definition:Standard Number Field|standard number fields]] $\Q$, $\R$ or $\C$.
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\map \MM n$ denote the [[Definition:Matrix Space|matri... | Consider the [[Definition:Algebraic Structure with Two Operations|algebraic structure]] $\struct {\map \MM {m, n}, +, \circ}$, where:
:$+$ denotes [[Definition:Matrix Entrywise Addition|matrix entrywise addition]]
:$\circ$ denotes [[Definition:Matrix Product (Conventional)|(conventional) matrix multiplication]].
From ... | Left and Right Inverses of Square Matrix over Field are Equal | https://proofwiki.org/wiki/Left_and_Right_Inverses_of_Square_Matrix_over_Field_are_Equal | https://proofwiki.org/wiki/Left_and_Right_Inverses_of_Square_Matrix_over_Field_are_Equal | [
"Inverse Matrices"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Standard Number Field",
"Definition:Strictly Positive/Integer",
"Definition:Matrix Space",
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix/Square Matrix",
"Definition:Inverse Matrix/Left",
"Definition:Inverse Matrix/Right",
"Definition:I... | [
"Definition:Algebraic Structure/Two Operations",
"Definition:Matrix Entrywise Addition",
"Definition:Matrix Product (Conventional)",
"Ring of Square Matrices over Field is Ring with Unity",
"Definition:Ring with Unity",
"Definition:A Fortiori",
"Definition:Monoid",
"Left Inverse and Right Inverse is I... |
proofwiki-17345 | Inverse of Square Matrix over Field is Unique | Let $\Bbb F$ be a field, usually one of the standard number fields $\Q$, $\R$ or $\C$.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\map \MM n$ denote the matrix space of order $n$ square matrices over $\Bbb F$.
Let $\mathbf B$ be an inverse matrix of $\mathbf A$.
Then $\mathbf B$ is the only inverse matr... | Consider the algebraic structure $\struct {\map \MM {m, n}, +, \circ}$, where:
:$+$ denotes matrix entrywise addition
:$\circ$ denotes (conventional) matrix multiplication.
From Ring of Square Matrices over Field is Ring with Unity, $\struct {\map \MM {m, n}, +, \circ}$ is a ring with unity.
Hence a fortiori $\struct {... | Let $\Bbb F$ be a [[Definition:Field (Abstract Algebra)|field]], usually one of the [[Definition:Standard Number Field|standard number fields]] $\Q$, $\R$ or $\C$.
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\map \MM n$ denote the [[Definition:Matrix Space|matri... | Consider the [[Definition:Algebraic Structure with Two Operations|algebraic structure]] $\struct {\map \MM {m, n}, +, \circ}$, where:
:$+$ denotes [[Definition:Matrix Entrywise Addition|matrix entrywise addition]]
:$\circ$ denotes [[Definition:Matrix Product (Conventional)|(conventional) matrix multiplication]].
From ... | Inverse of Square Matrix over Field is Unique | https://proofwiki.org/wiki/Inverse_of_Square_Matrix_over_Field_is_Unique | https://proofwiki.org/wiki/Inverse_of_Square_Matrix_over_Field_is_Unique | [
"Inverse Matrices"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Standard Number Field",
"Definition:Strictly Positive/Integer",
"Definition:Matrix Space",
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix/Square Matrix",
"Definition:Inverse Matrix",
"Definition:Unique",
"Definition:Inverse Matrix"
] | [
"Definition:Algebraic Structure/Two Operations",
"Definition:Matrix Entrywise Addition",
"Definition:Matrix Product (Conventional)",
"Ring of Square Matrices over Field is Ring with Unity",
"Definition:Ring with Unity",
"Definition:A Fortiori",
"Definition:Monoid",
"Inverse in Monoid is Unique"
] |
proofwiki-17346 | Rank of Empty Set is Zero | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\rho : \powerset S \to \Z$ be the rank function of $M$.
Then:
:$\map \rho \O = 0$ | By matroid axiom $(\text I 1)$:
:$\O$ is independent
From Rank of Independent Subset Equals Cardinality:
:$\map \rho \O = \size \O$
From Cardinality of Empty Set:
:$\size \O = 0$
The result follows.
{{qed}} | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\rho : \powerset S \to \Z$ be the [[Definition:Rank Function (Matroid)|rank function]] of $M$.
Then:
:$\map \rho \O = 0$ | By [[Axiom:Matroid Axioms|matroid axiom $(\text I 1)$]]:
:$\O$ is [[Definition:Independent Subset (Matroid)|independent]]
From [[Rank of Independent Subset Equals Cardinality]]:
:$\map \rho \O = \size \O$
From [[Cardinality of Empty Set]]:
:$\size \O = 0$
The result follows.
{{qed}} | Rank of Empty Set is Zero | https://proofwiki.org/wiki/Rank_of_Empty_Set_is_Zero | https://proofwiki.org/wiki/Rank_of_Empty_Set_is_Zero | [
"Matroid Rank Functions"
] | [
"Definition:Matroid",
"Definition:Rank Function (Matroid)"
] | [
"Axiom:Matroid Axioms",
"Definition:Matroid/Independent Set",
"Rank of Independent Subset Equals Cardinality",
"Cardinality of Empty Set"
] |
proofwiki-17347 | Rank Function is Increasing | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\rho: \powerset S \to \Z$ be the rank function of $M$.
Let $A, B \subseteq S$ be subsets of $S$ such that $A \subseteq B$.
Then:
:$\map \rho A \le \map \rho B$ | Now:
{{begin-eqn}}
{{eqn | l = \map \rho A
| r = \max \set {\size X : X \subseteq A \land X \in \mathscr I}
| c = {{Defof|Rank Function (Matroid)|Rank Function}}
}}
{{eqn | r = \max \set {\size X : X \in \powerset A \land X \in \mathscr I}
| c = {{Defof|Power Set}} of $\O$
}}
{{eqn | r = \max \set {\s... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\rho: \powerset S \to \Z$ be the [[Definition:Rank Function (Matroid)|rank function]] of $M$.
Let $A, B \subseteq S$ be [[Definition:Subset|subsets]] of $S$ such that $A \subseteq B$.
Then:
:$\map \rho A \le \map \rho B$ | Now:
{{begin-eqn}}
{{eqn | l = \map \rho A
| r = \max \set {\size X : X \subseteq A \land X \in \mathscr I}
| c = {{Defof|Rank Function (Matroid)|Rank Function}}
}}
{{eqn | r = \max \set {\size X : X \in \powerset A \land X \in \mathscr I}
| c = {{Defof|Power Set}} of $\O$
}}
{{eqn | r = \max \set {\s... | Rank Function is Increasing | https://proofwiki.org/wiki/Rank_Function_is_Increasing | https://proofwiki.org/wiki/Rank_Function_is_Increasing | [
"Matroid Rank Functions"
] | [
"Definition:Matroid",
"Definition:Rank Function (Matroid)",
"Definition:Subset"
] | [
"Power Set of Subset",
"Set Intersection Preserves Subsets",
"Max of Subfamily of Operands Less or Equal to Max"
] |
proofwiki-17348 | Bounds for Rank of Subset | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\rho: \powerset S \to \Z$ be the rank function of $M$.
Let $A \subseteq S$ be subset of $S$.
Then:
:$0 \le \map \rho A \le \size A$ | By definition of the rank function:
{{begin-eqn}}
{{eqn | l = \map \rho A
| r = \max \set {\size X : X \subseteq A \land X \in \mathscr I}
}}
{{end-eqn}}
From Cardinality of Subset of Finite Set:
:$\forall X \subseteq A : \size X \le \size A$
In particular:
:$\forall X \subseteq A : X \in \mathscr I$ then $\size ... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\rho: \powerset S \to \Z$ be the [[Definition:Rank Function (Matroid)|rank function]] of $M$.
Let $A \subseteq S$ be [[Definition:Subset|subset]] of $S$.
Then:
:$0 \le \map \rho A \le \size A$ | By definition of the [[Definition:Rank Function (Matroid)|rank function]]:
{{begin-eqn}}
{{eqn | l = \map \rho A
| r = \max \set {\size X : X \subseteq A \land X \in \mathscr I}
}}
{{end-eqn}}
From [[Cardinality of Subset of Finite Set]]:
:$\forall X \subseteq A : \size X \le \size A$
In particular:
:$\forall ... | Bounds for Rank of Subset | https://proofwiki.org/wiki/Bounds_for_Rank_of_Subset | https://proofwiki.org/wiki/Bounds_for_Rank_of_Subset | [
"Matroid Rank Functions"
] | [
"Definition:Matroid",
"Definition:Rank Function (Matroid)",
"Definition:Subset"
] | [
"Definition:Rank Function (Matroid)",
"Cardinality of Subset of Finite Set",
"Max Operation Yields Supremum of Parameters/General Case",
"Empty Set is Subset of All Sets",
"Cardinality of Empty Set",
"Axiom:Matroid Axioms",
"Max Operation Yields Supremum of Parameters/General Case"
] |
proofwiki-17349 | Inverse of Transpose of Matrix is Transpose of Inverse | Let $\mathbf A$ be a matrix over a field.
Let $\mathbf A^\intercal$ denote the transpose of $\mathbf A$.
Let $\mathbf A$ be an nonsingular matrix.
Then $\mathbf A^\intercal$ is also nonsingular and:
:$\paren {\mathbf A^\intercal}^{-1} = \paren {\mathbf A^{-1} }^\intercal$
where $\mathbf A^{-1}$ denotes the inverse of $... | We have:
{{begin-eqn}}
{{eqn | l = \paren {\mathbf A^{-1} }^\intercal \mathbf A^\intercal
| r = \paren {\mathbf A \mathbf A^{-1} }^\intercal
| c = Transpose of Matrix Product
}}
{{eqn | r = \mathbf I^\intercal
| c = {{Defof|Inverse Matrix}}: $\mathbf I$ denotes Unit Matrix
}}
{{eqn | r = \mathbf I
... | Let $\mathbf A$ be a [[Definition:Matrix|matrix]] over a [[Definition:Field (Abstract Algebra)|field]].
Let $\mathbf A^\intercal$ denote the [[Definition:Transpose of Matrix|transpose]] of $\mathbf A$.
Let $\mathbf A$ be an [[Definition:Nonsingular Matrix|nonsingular matrix]].
Then $\mathbf A^\intercal$ is also [[D... | We have:
{{begin-eqn}}
{{eqn | l = \paren {\mathbf A^{-1} }^\intercal \mathbf A^\intercal
| r = \paren {\mathbf A \mathbf A^{-1} }^\intercal
| c = [[Transpose of Matrix Product]]
}}
{{eqn | r = \mathbf I^\intercal
| c = {{Defof|Inverse Matrix}}: $\mathbf I$ denotes [[Definition:Unit Matrix|Unit Matri... | Inverse of Transpose of Matrix is Transpose of Inverse | https://proofwiki.org/wiki/Inverse_of_Transpose_of_Matrix_is_Transpose_of_Inverse | https://proofwiki.org/wiki/Inverse_of_Transpose_of_Matrix_is_Transpose_of_Inverse | [
"Transposes of Matrices",
"Inverse Matrices"
] | [
"Definition:Matrix",
"Definition:Field (Abstract Algebra)",
"Definition:Transpose of Matrix",
"Definition:Nonsingular Matrix",
"Definition:Nonsingular Matrix",
"Definition:Inverse Matrix"
] | [
"Transpose of Matrix Product",
"Definition:Unit Matrix",
"Definition:Inverse Matrix",
"Inverse of Square Matrix over Field is Unique"
] |
proofwiki-17350 | Max Operation Yields Supremum of Parameters/General Case | Let $x_1, x_2, \dotsc, x_n \in S$ for some $n \in \N_{>0}$.
Then:
:$\max \set {x_1, x_2, \dotsc, x_n} = \sup \set {x_1, x_2, \dotsc, x_n}$ | We will prove the result by induction on the number of operands $n$.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\max \set {x_1, x_2, \dotsc, x_n} = \sup \set {x_1, x_2, \dotsc, x_n}$ | Let $x_1, x_2, \dotsc, x_n \in S$ for some $n \in \N_{>0}$.
Then:
:$\max \set {x_1, x_2, \dotsc, x_n} = \sup \set {x_1, x_2, \dotsc, x_n}$ | We will prove the result by [[Principle of Mathematical Induction|induction]] on the [[Definition:Cardinality|number]] of [[Definition:Operand|operands]] $n$.
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\max \set {x_1, x_2, \dotsc, x_n} = \sup \set {x_1, x_2, \dotsc, x_n}$ | Max Operation Yields Supremum of Parameters/General Case | https://proofwiki.org/wiki/Max_Operation_Yields_Supremum_of_Parameters/General_Case | https://proofwiki.org/wiki/Max_Operation_Yields_Supremum_of_Parameters/General_Case | [
"Max Operation"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Cardinality",
"Definition:Operation/Operand",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-17351 | Max of Subfamily of Operands Less or Equal to Max | Let $\struct {S, \preceq}$ be a totally ordered set.
Let $x_1, x_2, \dotsc, x_n \in S$ for some $n \in \N_{>0}$.
Let $\set{k_1, k_2, \dotsc, k_m} \subseteq \set{1, 2, \dotsc, n}$
Then:
:$\max \set {x_{k_1}, x_{k_2}, \dotsc, x_{k_m}} \preceq \max \set {x_1, x_2, \dotsc, x_n}$
where:
:$\max$ denotes the max operation | From Max Operation Yields Supremum of Operands:
:$\max \set {x_{k_1}, x_{k_2}, \dotsc, x_{k_m}} = \sup \set {x_{k_1}, x_{k_2}, \dotsc, x_{k_m} }$
and
:$\max \set {x_1, x_2, \dotsc, x_n} = \sup \set {x_1, x_2, \dotsc, x_n}$
Since $\set {k_1, k_2, \dotsc, k_m} \subseteq \set {1, 2, \dotsc, n}$ then:
:$\set {x_{k_1}, x_{k... | Let $\struct {S, \preceq}$ be a [[Definition:Totally Ordered Set|totally ordered set]].
Let $x_1, x_2, \dotsc, x_n \in S$ for some $n \in \N_{>0}$.
Let $\set{k_1, k_2, \dotsc, k_m} \subseteq \set{1, 2, \dotsc, n}$
Then:
:$\max \set {x_{k_1}, x_{k_2}, \dotsc, x_{k_m}} \preceq \max \set {x_1, x_2, \dotsc, x_n}$
wher... | From [[Max Operation Yields Supremum of Operands]]:
:$\max \set {x_{k_1}, x_{k_2}, \dotsc, x_{k_m}} = \sup \set {x_{k_1}, x_{k_2}, \dotsc, x_{k_m} }$
and
:$\max \set {x_1, x_2, \dotsc, x_n} = \sup \set {x_1, x_2, \dotsc, x_n}$
Since $\set {k_1, k_2, \dotsc, k_m} \subseteq \set {1, 2, \dotsc, n}$ then:
:$\set {x_{k_1}... | Max of Subfamily of Operands Less or Equal to Max | https://proofwiki.org/wiki/Max_of_Subfamily_of_Operands_Less_or_Equal_to_Max | https://proofwiki.org/wiki/Max_of_Subfamily_of_Operands_Less_or_Equal_to_Max | [
"Max Operation"
] | [
"Definition:Totally Ordered Set",
"Definition:Max Operation"
] | [
"Max Operation Yields Supremum of Parameters/General Case",
"Supremum of Subset",
"Category:Max Operation"
] |
proofwiki-17352 | Elementary Matrix corresponding to Elementary Row Operation/Scale Row | Let $e$ be the elementary row operation acting on $\mathbf I$ as:
{{begin-axiom}}
{{axiom | n = \text {ERO} 1
| t = For some $\lambda \in K_{\ne 0}$, multiply row $k$ of $\mathbf I$ by $\lambda$
| m = r_k \to \lambda r_k
}}
{{end-axiom}} | By definition of the unit matrix:
:$I_{a b} = \delta_{a b}$
where:
:$I_{a b}$ denotes the element of $\mathbf I$ whose indices are $\tuple {a, b}$.
By definition, $\mathbf E$ is the square matrix of order $m$ formed by applying $e$ to the unit matrix $\mathbf I$.
That is, all elements of row $k$ of $\mathbf I$ are to b... | Let $e$ be the [[Definition:Elementary Row Operation|elementary row operation]] acting on $\mathbf I$ as:
{{begin-axiom}}
{{axiom | n = \text {ERO} 1
| t = For some $\lambda \in K_{\ne 0}$, [[Definition:Matrix Scalar Product|multiply]] [[Definition:Row of Matrix|row]] $k$ of $\mathbf I$ by $\lambda$
| ... | By definition of the [[Definition:Unit Matrix|unit matrix]]:
:$I_{a b} = \delta_{a b}$
where:
:$I_{a b}$ denotes the [[Definition:Element of Matrix|element]] of $\mathbf I$ whose [[Definition:Index of Matrix Element|indices]] are $\tuple {a, b}$.
By definition, $\mathbf E$ is the [[Definition:Square Matrix|square mat... | Elementary Matrix corresponding to Elementary Row Operation/Scale Row | https://proofwiki.org/wiki/Elementary_Matrix_corresponding_to_Elementary_Row_Operation/Scale_Row | https://proofwiki.org/wiki/Elementary_Matrix_corresponding_to_Elementary_Row_Operation/Scale_Row | [
"Elementary Matrix corresponding to Elementary Row Operation"
] | [
"Definition:Elementary Operation/Row",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Row"
] | [
"Definition:Unit Matrix",
"Definition:Matrix/Element",
"Definition:Matrix/Indices",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Unit Matrix",
"Definition:Matrix/Element",
"Definition:Matrix/Row",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:... |
proofwiki-17353 | Elementary Matrix corresponding to Elementary Row Operation/Scale Row and Add | Let $e$ be the elementary row operation acting on $\mathbf I$ as:
{{begin-axiom}}
{{axiom | n = \text {ERO} 2
| t = For some $\lambda \in K$, add $\lambda$ times row $j$ to row $i$
| m = r_i \to r_i + \lambda r_j
}}
{{end-axiom}} | By definition of the unit matrix:
:$I_{a b} = \delta_{a b}$
where:
:$I_{a b}$ denotes the element of $\mathbf I$ whose indices are $\tuple {a, b}$.
By definition, $\mathbf E$ is the square matrix of order $m$ formed by applying $e$ to the unit matrix $\mathbf I$.
That is, all elements of row $i$ of $\mathbf I$ are to h... | Let $e$ be the [[Definition:Elementary Row Operation|elementary row operation]] acting on $\mathbf I$ as:
{{begin-axiom}}
{{axiom | n = \text {ERO} 2
| t = For some $\lambda \in K$, add $\lambda$ [[Definition:Matrix Scalar Product|times]] [[Definition:Row of Matrix|row]] $j$ to [[Definition:Row of Matrix|row]]... | By definition of the [[Definition:Unit Matrix|unit matrix]]:
:$I_{a b} = \delta_{a b}$
where:
:$I_{a b}$ denotes the [[Definition:Element of Matrix|element]] of $\mathbf I$ whose [[Definition:Index of Matrix Element|indices]] are $\tuple {a, b}$.
By definition, $\mathbf E$ is the [[Definition:Square Matrix|square mat... | Elementary Matrix corresponding to Elementary Row Operation/Scale Row and Add | https://proofwiki.org/wiki/Elementary_Matrix_corresponding_to_Elementary_Row_Operation/Scale_Row_and_Add | https://proofwiki.org/wiki/Elementary_Matrix_corresponding_to_Elementary_Row_Operation/Scale_Row_and_Add | [
"Elementary Matrix corresponding to Elementary Row Operation"
] | [
"Definition:Elementary Operation/Row",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Row",
"Definition:Matrix/Row"
] | [
"Definition:Unit Matrix",
"Definition:Matrix/Element",
"Definition:Matrix/Indices",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Unit Matrix",
"Definition:Matrix/Element",
"Definition:Matrix/Row",
"Definition:Matrix/Element",
"Definition:Matrix/Row",
"D... |
proofwiki-17354 | Elementary Matrix corresponding to Elementary Row Operation/Exchange Rows | Let $e$ be the elementary row operation acting on $\mathbf I$ as:
{{begin-axiom}}
{{axiom | n = \text {ERO} 3
| t = Interchange rows $i$ and $j$
| m = r_i \leftrightarrow r_j
}}
{{end-axiom}} | By definition of the unit matrix:
:$I_{a b} = \delta_{a b}$
where:
:$I_{a b}$ denotes the element of $\mathbf I$ whose indices are $\tuple {a, b}$.
By definition, $\mathbf E$ is the square matrix of order $m$ formed by applying $e$ to the unit matrix $\mathbf I$.
That is, all elements of row $i$ of $\mathbf I$ are to b... | Let $e$ be the [[Definition:Elementary Row Operation|elementary row operation]] acting on $\mathbf I$ as:
{{begin-axiom}}
{{axiom | n = \text {ERO} 3
| t = Interchange [[Definition:Row of Matrix|rows]] $i$ and $j$
| m = r_i \leftrightarrow r_j
}}
{{end-axiom}} | By definition of the [[Definition:Unit Matrix|unit matrix]]:
:$I_{a b} = \delta_{a b}$
where:
:$I_{a b}$ denotes the [[Definition:Element of Matrix|element]] of $\mathbf I$ whose [[Definition:Index of Matrix Element|indices]] are $\tuple {a, b}$.
By definition, $\mathbf E$ is the [[Definition:Square Matrix|square mat... | Elementary Matrix corresponding to Elementary Row Operation/Exchange Rows | https://proofwiki.org/wiki/Elementary_Matrix_corresponding_to_Elementary_Row_Operation/Exchange_Rows | https://proofwiki.org/wiki/Elementary_Matrix_corresponding_to_Elementary_Row_Operation/Exchange_Rows | [
"Elementary Matrix corresponding to Elementary Row Operation"
] | [
"Definition:Elementary Operation/Row",
"Definition:Matrix/Row"
] | [
"Definition:Unit Matrix",
"Definition:Matrix/Element",
"Definition:Matrix/Indices",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Unit Matrix",
"Definition:Matrix/Element",
"Definition:Matrix/Row",
"Definition:Matrix/Element",
"Definition:Matrix/Row",
"D... |
proofwiki-17355 | Power Set of Singleton | Let $x$ be an object.
Then the power set of the singleton $\set x$ is:
:$\powerset {\set x} = \set {\O, \set x}$ | From Empty Set is Subset of All Sets:
:$\O \in \powerset {\set x}$
Let $A \in \powerset {\set x}$ such that $A \ne \O$
That is:
{{begin-eqn}}
{{eqn | r = A \subseteq \set x \land A \ne \O
| o =
| c =
}}
{{eqn | ll= \leadsto
| r = A \subseteq \set x \land \exists y : y \in A
| o =
| c = {{... | Let $x$ be an [[Definition:Object|object]].
Then the [[Definition:Power Set|power set]] of the [[Definition:Singleton|singleton]] $\set x$ is:
:$\powerset {\set x} = \set {\O, \set x}$ | From [[Empty Set is Subset of All Sets]]:
:$\O \in \powerset {\set x}$
Let $A \in \powerset {\set x}$ such that $A \ne \O$
That is:
{{begin-eqn}}
{{eqn | r = A \subseteq \set x \land A \ne \O
| o =
| c =
}}
{{eqn | ll= \leadsto
| r = A \subseteq \set x \land \exists y : y \in A
| o =
|... | Power Set of Singleton | https://proofwiki.org/wiki/Power_Set_of_Singleton | https://proofwiki.org/wiki/Power_Set_of_Singleton | [
"Power Set",
"Singletons"
] | [
"Definition:Object",
"Definition:Power Set",
"Definition:Singleton"
] | [
"Empty Set is Subset of All Sets",
"Singleton of Element is Subset",
"Definition:Subset",
"Category:Power Set",
"Category:Singletons"
] |
proofwiki-17356 | Row Operation to Clear First Column of Matrix | Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$.
Then there exists a row operation to convert $\mathbf A$ into another $m \times n$ matrix $\mathbf B = \sqbrk b_{m n}$ with the following properties:
:$(1): \quad$ Except possibly for element $b_{1 1}$, all the elements of column $1$ are $0$
:... | The following algorithm generates a sequence of elementary row operations which convert $\mathbf A$ to $\mathbf B$.
Let $\mathbf A' = \sqbrk {a'}_{m n}$ denote the state of $\mathbf A$ after having processed the latest step.
After each step, an implicit step can be included that requires that the form of $\mathbf A'$ i... | Let $\mathbf A = \sqbrk a_{m n}$ be an [[Definition:Matrix|$m \times n$ matrix]] over a [[Definition:Field (Abstract Algebra)|field]] $K$.
Then there exists a [[Definition:Row Operation|row operation]] to convert $\mathbf A$ into another [[Definition:Matrix|$m \times n$ matrix]] $\mathbf B = \sqbrk b_{m n}$ with the f... | The following [[Definition:Algorithm|algorithm]] generates a [[Definition:Sequence|sequence]] of [[Definition:Elementary Row Operation|elementary row operations]] which convert $\mathbf A$ to $\mathbf B$.
Let $\mathbf A' = \sqbrk {a'}_{m n}$ denote the state of $\mathbf A$ after having processed the latest step.
Afte... | Row Operation to Clear First Column of Matrix | https://proofwiki.org/wiki/Row_Operation_to_Clear_First_Column_of_Matrix | https://proofwiki.org/wiki/Row_Operation_to_Clear_First_Column_of_Matrix | [
"Row Operations",
"Row Operation to Clear First Column of Matrix"
] | [
"Definition:Matrix",
"Definition:Field (Abstract Algebra)",
"Definition:Row Operation",
"Definition:Matrix",
"Definition:Matrix/Element",
"Definition:Matrix/Element",
"Definition:Matrix/Column"
] | [
"Definition:Algorithm",
"Definition:Sequence",
"Definition:Elementary Operation/Row",
"Definition:Matrix/Element",
"Definition:Matrix/Column",
"Definition:Row Operation",
"Definition:Unit Matrix",
"Definition:Matrix/Element",
"Definition:Matrix/Row",
"Definition:Elementary Operation/Row",
"Defin... |
proofwiki-17357 | Matrix is Row Equivalent to Echelon Matrix | Let $\mathbf A = \sqbrk a_{m n}$ be a matrix of order $m \times n$ over a field $F$.
Then $A$ is row equivalent to an echelon matrix of order $m \times n$. | Using the operation Row Operation to Clear First Column of Matrix, $\mathbf A$ is converted to $\mathbf B$, which will be in the form:
:<nowiki>$\begin{bmatrix}
0 & \cdots & 0 & 1 & b_{1, j + 1} & \cdots & b_{1 n} \\
0 & \cdots & 0 & 0 & b_{2, j + 1} & \cdots & b_{2 n} \\
\vdots & \ddots &... | Let $\mathbf A = \sqbrk a_{m n}$ be a [[Definition:Matrix|matrix]] of [[Definition:Order of Matrix|order]] $m \times n$ over a [[Definition:Field (Abstract Algebra)|field]] $F$.
Then $A$ is [[Definition:Row Equivalence|row equivalent]] to an [[Definition:Echelon Matrix|echelon matrix]] of [[Definition:Order of Matrix... | Using the operation [[Row Operation to Clear First Column of Matrix]], $\mathbf A$ is converted to $\mathbf B$, which will be in the form:
:<nowiki>$\begin{bmatrix}
0 & \cdots & 0 & 1 & b_{1, j + 1} & \cdots & b_{1 n} \\
0 & \cdots & 0 & 0 & b_{2, j + 1} & \cdots & b_{2 n} \\
\vdots & \dd... | Matrix is Row Equivalent to Echelon Matrix | https://proofwiki.org/wiki/Matrix_is_Row_Equivalent_to_Echelon_Matrix | https://proofwiki.org/wiki/Matrix_is_Row_Equivalent_to_Echelon_Matrix | [
"Matrix is Row Equivalent to Echelon Matrix",
"Row Equivalence",
"Echelon Matrices"
] | [
"Definition:Matrix",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Row Equivalence",
"Definition:Echelon Matrix",
"Definition:Matrix/Order"
] | [
"Row Operation to Clear First Column of Matrix",
"Definition:Matrix/Zero Row or Column",
"Definition:Elementary Operation/Row",
"Definition:Submatrix",
"Definition:Submatrix",
"Definition:Submatrix",
"Definition:Matrix",
"Definition:Echelon Matrix/Echelon Form"
] |
proofwiki-17358 | Singleton is Independent implies Rank is One/Corollary | :$\set x$ is an independent subset {{iff}} $\map \rho {\set x} = 1$ | By definition of an independent subset:
:$x$ is an independent subset {{iff}} $\set x \notin \mathscr I$
From Singleton is Independent implies Rank is One:
:if $\set x \in \mathscr I$ then $\map \rho {\set x} = 1$
From Singleton is Dependent implies Rank is Zero:
:if $\set x \notin \mathscr I$ then $\map \rho {\set x} ... | :$\set x$ is an [[Definition:Independent Subset (Matroid)|independent subset]] {{iff}} $\map \rho {\set x} = 1$ | By definition of an [[Definition:Independent Subset (Matroid)|independent subset]]:
:$x$ is an [[Definition:Independent Subset (Matroid)|independent subset]] {{iff}} $\set x \notin \mathscr I$
From [[Singleton is Independent implies Rank is One]]:
:if $\set x \in \mathscr I$ then $\map \rho {\set x} = 1$
From [[Singl... | Singleton is Independent implies Rank is One/Corollary | https://proofwiki.org/wiki/Singleton_is_Independent_implies_Rank_is_One/Corollary | https://proofwiki.org/wiki/Singleton_is_Independent_implies_Rank_is_One/Corollary | [
"Matroid Independent Subsets",
"Matroid Rank Functions"
] | [
"Definition:Matroid/Independent Set"
] | [
"Definition:Matroid/Independent Set",
"Definition:Matroid/Independent Set",
"Singleton is Independent implies Rank is One",
"Singleton is Dependent implies Rank is Zero",
"Category:Matroid Independent Subsets",
"Category:Matroid Rank Functions"
] |
proofwiki-17359 | Singleton is Independent implies Rank is One | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $x \in S$.
Let $\set x$ be independent.
Then:
:$\map \rho {\set x} = 1$
where $\rho$ denotes the rank function of $M$. | From Rank of Independent Subset Equals Cardinality:
:$\map \rho {\set x} = \size {\set x}$
From Cardinality of Singleton:
:$\size {\set x} = 1$
The result follows.
{{qed}}
Category:Matroid Independent Subsets
Category:Matroid Rank Functions
mo4v8tljdjkn871iojpx49jz0d441ih | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $x \in S$.
Let $\set x$ be [[Definition:Independent Subset (Matroid)|independent]].
Then:
:$\map \rho {\set x} = 1$
where $\rho$ denotes the [[Definition:Rank Function (Matroid)|rank function]] of $M$. | From [[Rank of Independent Subset Equals Cardinality]]:
:$\map \rho {\set x} = \size {\set x}$
From [[Cardinality of Singleton]]:
:$\size {\set x} = 1$
The result follows.
{{qed}}
[[Category:Matroid Independent Subsets]]
[[Category:Matroid Rank Functions]]
mo4v8tljdjkn871iojpx49jz0d441ih | Singleton is Independent implies Rank is One | https://proofwiki.org/wiki/Singleton_is_Independent_implies_Rank_is_One | https://proofwiki.org/wiki/Singleton_is_Independent_implies_Rank_is_One | [
"Matroid Independent Subsets",
"Matroid Rank Functions"
] | [
"Definition:Matroid",
"Definition:Matroid/Independent Set",
"Definition:Rank Function (Matroid)"
] | [
"Rank of Independent Subset Equals Cardinality",
"Cardinality of Singleton",
"Category:Matroid Independent Subsets",
"Category:Matroid Rank Functions"
] |
proofwiki-17360 | Singleton is Dependent implies Rank is Zero | :$\map \rho {\set x} = 0$ | By definition of a dependent subset:
:$\set x \notin \mathscr I$
Then:
{{begin-eqn}}
{{eqn | l = \map \rho {\set x}
| r = \max \set{\size A : A \in \powerset {\set x} \land A \in \mathscr I}
| c = {{Defof|Rank Function (Matroid)|Rank Function}}
}}
{{eqn | r = \max \set {\size A : A \in \set {\O, \set x} \l... | :$\map \rho {\set x} = 0$ | By definition of a [[Definition:Dependent Subset (Matroid)|dependent subset]]:
:$\set x \notin \mathscr I$
Then:
{{begin-eqn}}
{{eqn | l = \map \rho {\set x}
| r = \max \set{\size A : A \in \powerset {\set x} \land A \in \mathscr I}
| c = {{Defof|Rank Function (Matroid)|Rank Function}}
}}
{{eqn | r = \max... | Singleton is Dependent implies Rank is Zero | https://proofwiki.org/wiki/Singleton_is_Dependent_implies_Rank_is_Zero | https://proofwiki.org/wiki/Singleton_is_Dependent_implies_Rank_is_Zero | [
"Matroid Dependent Subsets",
"Matroid Rank Functions"
] | [] | [
"Definition:Matroid/Dependent Set",
"Power Set of Singleton",
"Axiom:Matroid Axioms",
"Cardinality of Empty Set",
"Category:Matroid Dependent Subsets",
"Category:Matroid Rank Functions"
] |
proofwiki-17361 | System of Simultaneous Equations may have No Solution | Let $S$ be a system of simultaneous equations.
Then it is possible that $S$ may have a solution set which is empty. | Consider this system of simultaneous linear equations:
{{begin-eqn}}
{{eqn | n = 1
| l = x_1 + x_2
| r = 2
}}
{{eqn | n = 2
| l = 2 x_1 + 2 x_2
| r = 3
}}
{{end-eqn}}
From its evaluation it is seen to have no solutions.
Hence the result.
{{qed}} | Let $S$ be a [[Definition:Simultaneous Equations|system of simultaneous equations]].
Then it is possible that $S$ may have a [[Definition:Solution Set to System of Simultaneous Equations|solution set]] which is [[Definition:Empty Set|empty]]. | Consider this [[Simultaneous Linear Equations/Examples/Arbitrary System 2|system of simultaneous linear equations]]:
{{begin-eqn}}
{{eqn | n = 1
| l = x_1 + x_2
| r = 2
}}
{{eqn | n = 2
| l = 2 x_1 + 2 x_2
| r = 3
}}
{{end-eqn}}
From its [[Simultaneous Linear Equations/Examples/Arbitrary Syste... | System of Simultaneous Equations may have No Solution | https://proofwiki.org/wiki/System_of_Simultaneous_Equations_may_have_No_Solution | https://proofwiki.org/wiki/System_of_Simultaneous_Equations_may_have_No_Solution | [
"Simultaneous Equations"
] | [
"Definition:Simultaneous Equations",
"Definition:Simultaneous Equations/Solution Set",
"Definition:Empty Set"
] | [
"Simultaneous Linear Equations/Examples/Arbitrary System 2",
"Simultaneous Linear Equations/Examples/Arbitrary System 2",
"Definition:Simultaneous Equations/Solution"
] |
proofwiki-17362 | System of Simultaneous Equations may have Unique Solution | Let $S$ be a system of simultaneous equations.
Then it is possible that $S$ may have a solution set which is a singleton. | Consider this system of simultaneous linear equations:
{{begin-eqn}}
{{eqn | n = 1
| l = x_1 - 2 x_2 + x_3
| r = 1
}}
{{eqn | n = 2
| l = 2 x_1 - x_2 + x_3
| r = 2
}}
{{eqn | n = 3
| l = 4 x_1 + x_2 - x_3
| r = 1
}}
{{end-eqn}}
From its evaluation it has the following unique solution... | Let $S$ be a [[Definition:Simultaneous Equations|system of simultaneous equations]].
Then it is possible that $S$ may have a [[Definition:Solution Set to System of Simultaneous Equations|solution set]] which is a [[Definition:Singleton|singleton]]. | Consider this [[Simultaneous Linear Equations/Examples/Arbitrary System 1|system of simultaneous linear equations]]:
{{begin-eqn}}
{{eqn | n = 1
| l = x_1 - 2 x_2 + x_3
| r = 1
}}
{{eqn | n = 2
| l = 2 x_1 - x_2 + x_3
| r = 2
}}
{{eqn | n = 3
| l = 4 x_1 + x_2 - x_3
| r = 1
}}
{{end... | System of Simultaneous Equations may have Unique Solution | https://proofwiki.org/wiki/System_of_Simultaneous_Equations_may_have_Unique_Solution | https://proofwiki.org/wiki/System_of_Simultaneous_Equations_may_have_Unique_Solution | [
"Simultaneous Equations"
] | [
"Definition:Simultaneous Equations",
"Definition:Simultaneous Equations/Solution Set",
"Definition:Singleton"
] | [
"Simultaneous Linear Equations/Examples/Arbitrary System 1",
"Simultaneous Linear Equations/Examples/Arbitrary System 1",
"Definition:Unique",
"Definition:Simultaneous Equations/Solution"
] |
proofwiki-17363 | System of Simultaneous Equations may have Multiple Solutions | Let $S$ be a system of simultaneous equations.
Then it is possible that $S$ may have a solution set which is a singleton. | Consider this system of simultaneous linear equations:
{{begin-eqn}}
{{eqn | n = 1
| l = x_1 - 2 x_2 + x_3
| r = 1
}}
{{eqn | n = 2
| l = 2 x_1 - x_2 + x_3
| r = 2
}}
{{end-eqn}}
From its evaluation it has the following solutions:
{{begin-eqn}}
{{eqn | l = x_1
| r = 1 - \dfrac t 3
}}
{{eqn... | Let $S$ be a [[Definition:Simultaneous Equations|system of simultaneous equations]].
Then it is possible that $S$ may have a [[Definition:Solution Set to System of Simultaneous Equations|solution set]] which is a [[Definition:Singleton|singleton]]. | Consider this [[Simultaneous Linear Equations/Examples/Arbitrary System 1|system of simultaneous linear equations]]:
{{begin-eqn}}
{{eqn | n = 1
| l = x_1 - 2 x_2 + x_3
| r = 1
}}
{{eqn | n = 2
| l = 2 x_1 - x_2 + x_3
| r = 2
}}
{{end-eqn}}
From its [[Simultaneous Linear Equations/Examples/Arb... | System of Simultaneous Equations may have Multiple Solutions | https://proofwiki.org/wiki/System_of_Simultaneous_Equations_may_have_Multiple_Solutions | https://proofwiki.org/wiki/System_of_Simultaneous_Equations_may_have_Multiple_Solutions | [
"Simultaneous Equations"
] | [
"Definition:Simultaneous Equations",
"Definition:Simultaneous Equations/Solution Set",
"Definition:Singleton"
] | [
"Simultaneous Linear Equations/Examples/Arbitrary System 1",
"Simultaneous Linear Equations/Examples/Arbitrary System 2",
"Definition:Simultaneous Equations/Solution",
"Definition:Number",
"Definition:Simultaneous Equations/Solution",
"Definition:Cardinality",
"Definition:Variable/Domain"
] |
proofwiki-17364 | Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach | Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Then $X$ is a Banach space {{iff}}:
:every absolutely convergent series $\ds \sum_{n \mathop = 1}^\infty a_n$ is convergent. | === Necessary Condition ===
{{:Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach/Necessary Condition}}{{qed|lemma}} | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Then $X$ is a [[Definition:Banach Space|Banach space]] {{iff}}:
:every [[Definition:Absolutely Convergent Series|absolutely convergent series]] $\ds \sum_{n \mathop = 1}^\infty a_n$ is [[Definition:Convergent Series|c... | === [[Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach/Necessary Condition|Necessary Condition]] ===
{{:Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach/Necessary Condition}}{{qed|lemma}} | Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach | https://proofwiki.org/wiki/Absolutely_Convergent_Series_in_Normed_Vector_Space_is_Convergent_iff_Space_is_Banach | https://proofwiki.org/wiki/Absolutely_Convergent_Series_in_Normed_Vector_Space_is_Convergent_iff_Space_is_Banach | [
"Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach",
"Absolute Convergence",
"Banach Spaces",
"Absolutely Convergent Series is Convergent",
"Convergent Sequences (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Banach Space",
"Definition:Absolutely Convergent Series",
"Definition:Convergent Series"
] | [
"Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach/Necessary Condition"
] |
proofwiki-17365 | Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach/Necessary Condition | Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $X$.
Suppose $X$ is a Banach space.
Then $\ds \sum_{n \mathop = 1}^\infty a_n$ is convergent. | That $\ds \sum_{n \mathop = 1}^\infty a_n$ is absolutely convergent means that $\ds \sum_{n \mathop = 1}^\infty \norm {a_n}$ converges in $\R$.
Hence by Convergent Sequence in Normed Vector Space is Cauchy Sequence:
:the sequence of partial sums is a Cauchy sequence.
Now let $\epsilon > 0$.
Let $N \in \N$ such that for... | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be an [[Definition:Absolutely Convergent Series|absolutely convergent series]] in $X$.
Suppose $X$ is a [[Definition:Banach Space|Banach space]].
Then $\ds \sum_{n \mathop =... | That $\ds \sum_{n \mathop = 1}^\infty a_n$ is [[Definition:Absolutely Convergent Series|absolutely convergent]] means that $\ds \sum_{n \mathop = 1}^\infty \norm {a_n}$ [[Definition:Convergent Series|converges]] in $\R$.
Hence by [[Convergent Sequence in Normed Vector Space is Cauchy Sequence]]:
:the [[Definition:Sequ... | Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach/Necessary Condition | https://proofwiki.org/wiki/Absolutely_Convergent_Series_in_Normed_Vector_Space_is_Convergent_iff_Space_is_Banach/Necessary_Condition | https://proofwiki.org/wiki/Absolutely_Convergent_Series_in_Normed_Vector_Space_is_Convergent_iff_Space_is_Banach/Necessary_Condition | [
"Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach"
] | [
"Definition:Normed Vector Space",
"Definition:Absolutely Convergent Series",
"Definition:Banach Space",
"Definition:Convergent Series"
] | [
"Definition:Absolutely Convergent Series",
"Definition:Convergent Series",
"Convergent Sequence is Cauchy Sequence/Normed Vector Space",
"Definition:Series/Sequence of Partial Sums",
"Definition:Cauchy Sequence",
"Definition:Cauchy Sequence",
"Definition:Norm/Vector Space",
"Definition:Series/Sequence... |
proofwiki-17366 | Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach/Sufficient Condition | Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $X$.
Suppose $\ds \sum_{n \mathop = 1}^\infty a_n$ is convergent.
Then $X$ is a Banach space. | Let $\sequence {x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $X$.
We have that:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$
We will prove the existence of a subsequence $\sequence {x_{n_k} }_{k \mathop \in \N}$ such that:
:$n > n_k \implies \norm ... | Let $\struct {X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be an [[Definition:Absolutely Convergent Series|absolutely convergent series]] in $X$.
Suppose $\ds \sum_{n \mathop = 1}^\infty a_n$ is [[Definition:Convergent Series|convergen... | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Cauchy Sequence in Normed Vector Space|Cauchy sequence]] in $X$.
We have that:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$
We will prove the existence of a [[Definition:Subsequence|subseque... | Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach/Sufficient Condition | https://proofwiki.org/wiki/Absolutely_Convergent_Series_in_Normed_Vector_Space_is_Convergent_iff_Space_is_Banach/Sufficient_Condition | https://proofwiki.org/wiki/Absolutely_Convergent_Series_in_Normed_Vector_Space_is_Convergent_iff_Space_is_Banach/Sufficient_Condition | [
"Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach"
] | [
"Definition:Normed Vector Space",
"Definition:Absolutely Convergent Series",
"Definition:Convergent Series",
"Definition:Banach Space"
] | [
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Subsequence",
"Definition:Sequence",
"Definition:Series",
"Definition:Absolutely Convergent Series",
"Definition:Convergent Series/Normed Vector Space/Definition 2",
"Definition:Telescoping Series",
"Definition:Convergent Series/Normed Vect... |
proofwiki-17367 | Sine of Integer Multiple of Argument/Formulation 2 | {{begin-eqn}}
{{eqn | l = \sin n \theta
| r = \cos^n \theta \paren {\dbinom n 1 \paren {\tan \theta} - \dbinom n 3 \paren {\tan \theta}^3 + \dbinom n 5 \paren {\tan \theta}^5 - \cdots}
| c =
}}
{{eqn | r = \cos^n \theta \sum_{k \mathop \ge 0} \paren {-1}^k \dbinom n {2 k + 1} \paren {\tan^{2 k + 1} \theta}... | By De Moivre's Formula:
:$\cos n \theta + i \sin n \theta = \paren {\cos \theta + i \sin \theta}^n$
As $n \in \Z_{>0}$, we use the Binomial Theorem on the {{RHS}}, resulting in:
:$\ds \cos n \theta + i \sin n \theta = \sum_{k \mathop \ge 0} \binom n k \paren {\cos^{n - k} \theta} \paren {i \sin \theta}^k$
When $k$ is o... | {{begin-eqn}}
{{eqn | l = \sin n \theta
| r = \cos^n \theta \paren {\dbinom n 1 \paren {\tan \theta} - \dbinom n 3 \paren {\tan \theta}^3 + \dbinom n 5 \paren {\tan \theta}^5 - \cdots}
| c =
}}
{{eqn | r = \cos^n \theta \sum_{k \mathop \ge 0} \paren {-1}^k \dbinom n {2 k + 1} \paren {\tan^{2 k + 1} \theta}... | By [[De Moivre's Formula]]:
:$\cos n \theta + i \sin n \theta = \paren {\cos \theta + i \sin \theta}^n$
As $n \in \Z_{>0}$, we use the [[Binomial Theorem]] on the {{RHS}}, resulting in:
:$\ds \cos n \theta + i \sin n \theta = \sum_{k \mathop \ge 0} \binom n k \paren {\cos^{n - k} \theta} \paren {i \sin \theta}^k$
Wh... | Sine of Integer Multiple of Argument/Formulation 2 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_2 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_2 | [
"Sine of Integer Multiple of Argument"
] | [] | [
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Odd Integer",
"Definition:Imaginary Number",
"Definition:Complex Number/Imaginary Part",
"Definition:Odd Integer"
] |
proofwiki-17368 | Trivial Solution to System of Homogeneous Simultaneous Linear Equations is Solution | Let $S$ be a '''system of homogeneous simultaneous linear equations''':
:$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$
Consider the trivial solution to $A$:
:$\tuple {x_1, x_2, \ldots, x_n}$
such that:
:$\forall j \in \set {1, 2, \ldots, n}: x_j = 0$
Then the trivial solution i... | Let $i \in \set {1, 2, \ldots, m}$.
We have:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n \alpha_{i j} x_j
| r = \sum_{j \mathop = 1}^n \alpha_{i j} \times 0
| c =
}}
{{eqn | r = \sum_{j \mathop = 1}^n 0
| c =
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
This holds for all $i \in \set {1, 2, \ld... | Let $S$ be a '''system of [[Definition:Homogeneous Simultaneous Linear Equations|homogeneous simultaneous linear equations]]''':
:$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$
Consider the [[Definition:Trivial Solution to Homogeneous Simultaneous Linear Equations|trivial solu... | Let $i \in \set {1, 2, \ldots, m}$.
We have:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n \alpha_{i j} x_j
| r = \sum_{j \mathop = 1}^n \alpha_{i j} \times 0
| c =
}}
{{eqn | r = \sum_{j \mathop = 1}^n 0
| c =
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
This holds for all $i \in \set {1, 2, ... | Trivial Solution to System of Homogeneous Simultaneous Linear Equations is Solution | https://proofwiki.org/wiki/Trivial_Solution_to_System_of_Homogeneous_Simultaneous_Linear_Equations_is_Solution | https://proofwiki.org/wiki/Trivial_Solution_to_System_of_Homogeneous_Simultaneous_Linear_Equations_is_Solution | [
"Simultaneous Linear Equations"
] | [
"Definition:Homogeneous Simultaneous Linear Equations",
"Definition:Trivial Solution to Homogeneous Simultaneous Linear Equations",
"Definition:Trivial Solution to Homogeneous Simultaneous Linear Equations",
"Definition:Simultaneous Equations/Linear Equations/Solution"
] | [
"Category:Simultaneous Linear Equations"
] |
proofwiki-17369 | Sine of Integer Multiple of Argument/Formulation 3 | {{begin-eqn}}
{{eqn | l = \sin n \theta
| r = \sin \theta \cos^{n - 1} \theta \paren {1 + 1 + \frac {\cos 2 \theta} {\cos^2 \theta} + \frac {\cos 3 \theta} {\cos^3 \theta} + \cdots + \frac {\cos \paren {n - 1} \theta} {\cos^{n - 1} \theta} }
| c =
}}
{{eqn | r = \sin \theta \cos^{n - 1} \theta \sum_{k \mat... | The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\ds \sin n \theta = \sin \theta \cos^{n - 1} \theta \sum_{k \mathop \ge 0} \frac {\cos k \theta} {\cos^k \theta}$ | {{begin-eqn}}
{{eqn | l = \sin n \theta
| r = \sin \theta \cos^{n - 1} \theta \paren {1 + 1 + \frac {\cos 2 \theta} {\cos^2 \theta} + \frac {\cos 3 \theta} {\cos^3 \theta} + \cdots + \frac {\cos \paren {n - 1} \theta} {\cos^{n - 1} \theta} }
| c =
}}
{{eqn | r = \sin \theta \cos^{n - 1} \theta \sum_{k \mat... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sin n \theta = \sin \theta \cos^{n - 1} \theta \sum_{k \mathop \ge 0} \frac {\cos k \theta} {\cos^k \theta}$ | Sine of Integer Multiple of Argument/Formulation 3 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_3 | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_3 | [
"Sine of Integer Multiple of Argument"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-17370 | Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations | Let $S$ be a system of simultaneous linear equations:
:$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
Let $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ denote the augmented matrix of $S$.
Let $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$ be obtained f... | We have that an elementary row operation $e$ is used to transform $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ to $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$.
Now, whatever $e$ is, $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$ is the augmented matrix of a system of simultaneous line... | Let $S$ be a system of [[Definition:Simultaneous Linear Equations|simultaneous linear equations]]:
:$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
Let $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ denote the [[Definition:Augmented Matrix of Simultaneous Linear ... | We have that an [[Definition:Elementary Row Operation|elementary row operation]] $e$ is used to transform $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ to $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$.
Now, whatever $e$ is, $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$ is the [[Defin... | Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations | https://proofwiki.org/wiki/Elementary_Row_Operation_on_Augmented_Matrix_leads_to_Equivalent_System_of_Simultaneous_Linear_Equations | https://proofwiki.org/wiki/Elementary_Row_Operation_on_Augmented_Matrix_leads_to_Equivalent_System_of_Simultaneous_Linear_Equations | [
"Elementary Row Operations",
"Simultaneous Linear Equations",
"Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations"
] | [
"Definition:Simultaneous Equations/Linear Equations",
"Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matrix",
"Definition:Elementary Operation/Row",
"Definition:Simultaneous Equations/Linear Equations",
"Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matr... | [
"Definition:Elementary Operation/Row",
"Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matrix",
"Definition:Simultaneous Equations/Linear Equations",
"Definition:Elementary Operation/Row",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Elementary Operation/Row",
... |
proofwiki-17371 | Existence of Inverse Elementary Row Operation | Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$.
Let $\mathbf A \in \map \MM {m, n}$ be a matrix.
Let $\map e {\mathbf A}$ be an elementary row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$.
Let $\map {e'} {\mathbf A'}$ be the inverse of $e$.
The... | Let us take each type of elementary row operation in turn.
For each $\map e {\mathbf A}$, we will construct $\map {e'} {\mathbf A'}$ which will transform $\mathbf A'$ into a new matrix $\mathbf A' ' \in \map \MM {m, n}$, which will then be demonstrated to equal $\mathbf A$.
In the below, let:
:$r_k$ denote row $k$ of $... | Let $\map \MM {m, n}$ be a [[Definition:Metric Space|metric space]] of [[Definition:Order of Matrix|order]] $m \times n$ over a [[Definition:Field (Abstract Algebra)|field]] $K$.
Let $\mathbf A \in \map \MM {m, n}$ be a [[Definition:Matrix|matrix]].
Let $\map e {\mathbf A}$ be an [[Definition:Elementary Row Operatio... | Let us take each type of [[Definition:Elementary Row Operation|elementary row operation]] in turn.
For each $\map e {\mathbf A}$, we will construct $\map {e'} {\mathbf A'}$ which will transform $\mathbf A'$ into a new [[Definition:Matrix|matrix]] $\mathbf A' ' \in \map \MM {m, n}$, which will then be demonstrated to ... | Existence of Inverse Elementary Row Operation | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Row_Operation | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Row_Operation | [
"Elementary Row Operations",
"Existence of Inverse Elementary Row Operation"
] | [
"Definition:Metric Space",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Matrix",
"Definition:Elementary Operation/Row",
"Definition:Matrix",
"Definition:Inverse of Elementary Row Operation",
"Definition:Elementary Operation/Row",
"Definition:Unique"
] | [
"Definition:Elementary Operation/Row",
"Definition:Matrix",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Elementary Operation/Row",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Matrix/Row... |
proofwiki-17372 | Superset of Dependent Set is Dependent | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $A, B \subseteq S$ such that $A \subseteq B$
If $A$ is a dependent subset then $B$ is a dependent subset. | From the contrapositive statement of matroid axiom $(\text I 2)$:
:$A \notin \mathscr I \implies B \notin \mathscr I$
By the definition of a dependent subset:
:If $A$ is not an dependent subset then $B$ is not an dependent subset.
{{qed}}
Category:Matroid Dependent Subsets
ed2sk63don3viuma9rtyxgx1bx7b39f | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $A, B \subseteq S$ such that $A \subseteq B$
If $A$ is a [[Definition:Dependent Subset (Matroid)|dependent subset]] then $B$ is a [[Definition:Dependent Subset (Matroid)|dependent subset]]. | From the [[Definition:Contrapositive Statement|contrapositive statement]] of [[Axiom:Matroid Axioms|matroid axiom $(\text I 2)$]]:
:$A \notin \mathscr I \implies B \notin \mathscr I$
By the definition of a [[Definition:Dependent Subset (Matroid)|dependent subset]]:
:If $A$ is not an [[Definition:Dependent Subset (Matr... | Superset of Dependent Set is Dependent | https://proofwiki.org/wiki/Superset_of_Dependent_Set_is_Dependent | https://proofwiki.org/wiki/Superset_of_Dependent_Set_is_Dependent | [
"Matroid Dependent Subsets"
] | [
"Definition:Matroid",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Dependent Set"
] | [
"Definition:Contrapositive Statement",
"Axiom:Matroid Axioms",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Dependent Set",
"Category:Matroid Dependent Subsets"
] |
proofwiki-17373 | Powers of 16 Modulo 20 | Let $n \in \Z_{> 0}$ be a strictly positive integer.
Then:
:$16^n \equiv 16 \pmod {20}$ | Proof by induction:
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$16^n \equiv 16 \pmod {20}$
=== Basis for the Induction ===
$\map P 1$ is the case:
:$16^1 \equiv 16 \pmod {20}$
Thus $\map P 1$ is seen to hold.
This is the basis for the induction.
=== Induction Hypothesis ===
Now it needs to be shown t... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Then:
:$16^n \equiv 16 \pmod {20}$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \Z_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$16^n \equiv 16 \pmod {20}$
=== Basis for the Induction ===
$\map P 1$ is the case:
:$16^1 \equiv 16 \pmod {20}$
Thus $\map P 1$ is seen to hold.
This is the [[Defi... | Powers of 16 Modulo 20/Proof 1 | https://proofwiki.org/wiki/Powers_of_16_Modulo_20 | https://proofwiki.org/wiki/Powers_of_16_Modulo_20/Proof_1 | [
"Powers of 16",
"Modulo Arithmetic",
"Powers of 16 Modulo 20"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Powers of 16 Modulo 20/Proof 1",
"Principle of Mathematical Induction"
] |
proofwiki-17374 | Powers of 16 Modulo 20 | Let $n \in \Z_{> 0}$ be a strictly positive integer.
Then:
:$16^n \equiv 16 \pmod {20}$ | {{begin-eqn}}
{{eqn | l = 16
| o = \equiv
| r = 16
| rr= \pmod {20}
}}
{{eqn | ll= \leadsto
| l = 16
| o = \equiv
| r = 0
| rr= \pmod 4
}}
{{eqn | lo= \text {and}
| l = 16
| o = \equiv
| r = 1
| rr= \pmod 5
}}
{{eqn | ll= \leadsto
| l = 16^n
... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Then:
:$16^n \equiv 16 \pmod {20}$ | {{begin-eqn}}
{{eqn | l = 16
| o = \equiv
| r = 16
| rr= \pmod {20}
}}
{{eqn | ll= \leadsto
| l = 16
| o = \equiv
| r = 0
| rr= \pmod 4
}}
{{eqn | lo= \text {and}
| l = 16
| o = \equiv
| r = 1
| rr= \pmod 5
}}
{{eqn | ll= \leadsto
| l = 16^n
... | Powers of 16 Modulo 20/Proof 2 | https://proofwiki.org/wiki/Powers_of_16_Modulo_20 | https://proofwiki.org/wiki/Powers_of_16_Modulo_20/Proof_2 | [
"Powers of 16",
"Modulo Arithmetic",
"Powers of 16 Modulo 20"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Chinese Remainder Theorem"
] |
proofwiki-17375 | Sine of Integer Multiple of Argument/Formulation 1/Lemma | :For $n \in \Z$:
{{begin-eqn}}
{{eqn | l = \map \cos {n \theta} \map \sin {\theta}
| r = \map \sin {n \theta} \map \cos {\theta} - \map \sin {\paren {n - 1 } \theta}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map \cos {n \theta} \map \sin {\theta}
| r = \map \cos {n \theta} \map \sin {\theta}
}}
{{eqn | r = \paren {\map \sin {n \theta} \map \cos {\theta} - \map \sin {n \theta} \map \cos {\theta} } + \map \cos {n \theta} \map \sin {\theta}
| c = add zero
}}
{{eqn | r = \map \sin {n \... | :For $n \in \Z$:
{{begin-eqn}}
{{eqn | l = \map \cos {n \theta} \map \sin {\theta}
| r = \map \sin {n \theta} \map \cos {\theta} - \map \sin {\paren {n - 1 } \theta}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map \cos {n \theta} \map \sin {\theta}
| r = \map \cos {n \theta} \map \sin {\theta}
}}
{{eqn | r = \paren {\map \sin {n \theta} \map \cos {\theta} - \map \sin {n \theta} \map \cos {\theta} } + \map \cos {n \theta} \map \sin {\theta}
| c = add zero
}}
{{eqn | r = \map \sin {n \... | Sine of Integer Multiple of Argument/Formulation 1/Lemma | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_1/Lemma | https://proofwiki.org/wiki/Sine_of_Integer_Multiple_of_Argument/Formulation_1/Lemma | [
"Sine of Integer Multiple of Argument"
] | [] | [
"Sine of Difference",
"Category:Sine of Integer Multiple of Argument"
] |
proofwiki-17376 | Conjugacy Class of Identity is only Conjugacy Class which is Subgroup | Let $G$ be a group.
Let $e$ denote the identity of $G$.
Let $\conjclass g$ denote the conjugacy class of the element $g$.
Then conjugacy class of identity is the only conjugacy class which is a subgroup of $G$:
:$\conjclass g < G \iff g = e$ | === Necessary Condition ===
Assume $g = e$.
Then by Identity of Group is in Singleton Conjugacy Class, $\conjclass e = \set e$, which is the trivial subgroup.
{{qed|lemma}} | Let $G$ be a [[Definition:Group|group]].
Let $e$ denote the [[Definition:Identity Element|identity]] of $G$.
Let $\conjclass g$ denote the [[Definition:Conjugacy Class|conjugacy class]] of the element $g$.
Then conjugacy class of identity is the only conjugacy class which is a [[Definition:Subgroup|subgroup]] of $G... | === Necessary Condition ===
Assume $g = e$.
Then by [[Identity of Group is in Singleton Conjugacy Class]], $\conjclass e = \set e$, which is the [[Definition:Trivial Subgroup|trivial subgroup]].
{{qed|lemma}} | Conjugacy Class of Identity is only Conjugacy Class which is Subgroup | https://proofwiki.org/wiki/Conjugacy_Class_of_Identity_is_only_Conjugacy_Class_which_is_Subgroup | https://proofwiki.org/wiki/Conjugacy_Class_of_Identity_is_only_Conjugacy_Class_which_is_Subgroup | [
"Conjugacy Classes"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Conjugacy Class",
"Definition:Subgroup"
] | [
"Identity of Group is in Singleton Conjugacy Class",
"Definition:Trivial Subgroup"
] |
proofwiki-17377 | Empty Group Word is Reduced | Let $S$ be a set
Let $\epsilon$ be the empty group word on $S$.
Then $\epsilon$ is reduced. | By definition, a group word $w = w_1 \cdots w_i \cdots w_n$ is reduced {{iff}}:
:$w_i \ne {w_{i + 1} }^{-1}$ for all $i \in \set {1, \ldots, n - 1}$
where $w_1, w_2, \ldots$ are elements of $S$.
We have {{hypothesis}} that $\epsilon$ is the empty group word on $S$.
Hence by definition it has no elements of $S$ in it.
H... | Let $S$ be a [[Definition:Set|set]]
Let $\epsilon$ be the [[Definition:Empty Group Word|empty group word]] on $S$.
Then $\epsilon$ is [[Definition:Reduced Group Word on Set|reduced]]. | By definition, a [[Definition:Group Word on Set|group word]] $w = w_1 \cdots w_i \cdots w_n$ is [[Definition:Reduced Group Word on Set|reduced]] {{iff}}:
:$w_i \ne {w_{i + 1} }^{-1}$ for all $i \in \set {1, \ldots, n - 1}$
where $w_1, w_2, \ldots$ are [[Definition:Element|elements]] of $S$.
We have {{hypothesis}} that... | Empty Group Word is Reduced | https://proofwiki.org/wiki/Empty_Group_Word_is_Reduced | https://proofwiki.org/wiki/Empty_Group_Word_is_Reduced | [
"Group Words"
] | [
"Definition:Set",
"Definition:Empty Group Word",
"Definition:Reduced Group Word on Set"
] | [
"Definition:Group Word on Set",
"Definition:Reduced Group Word on Set",
"Definition:Element",
"Definition:Empty Group Word",
"Definition:Element",
"Definition:Reduced Group Word on Set",
"Definition:Vacuous Truth",
"Category:Group Words"
] |
proofwiki-17378 | Scalar Multiplication Corresponds to Multiplication by 1x1 Matrix | Let $\map \MM 1$ denote the matrix space of square matrices of order $1$.
Let $\map \MM {1, n}$ denote the matrix space of order $1 \times n$.
Let $\mathbf A = \begin {pmatrix} a \end {pmatrix} \in \map \MM 1$ and $\mathbf B = \begin {pmatrix} b_1 & b_2 & \cdots & b_n \end{pmatrix} \in \map \MM {1, n}$.
Let $\mathbf C ... | By definition of (conventional) matrix product, $\mathbf C$ is of order $1 \times n$.
By definition of matrix scalar product, $\mathbf D$ is also of order $1 \times n$.
Consider arbitrary elements $c_i \in \mathbf C$ and $d_i \in \mathbf D$ for some index $i$ where $1 \le i \le n$.
We have:
{{begin-eqn}}
{{eqn | l = c_... | Let $\map \MM 1$ denote the [[Definition:Matrix Space|matrix space]] of [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order]] $1$.
Let $\map \MM {1, n}$ denote the [[Definition:Matrix Space|matrix space]] of [[Definition:Order of Square Matrix|order]] $1 \times n$.
Let $\mathbf A... | By definition of [[Definition:Matrix Product (Conventional)|(conventional) matrix product]], $\mathbf C$ is of [[Definition:Order of Matrix|order]] $1 \times n$.
By definition of [[Definition:Matrix Scalar Product|matrix scalar product]], $\mathbf D$ is also of [[Definition:Order of Matrix|order]] $1 \times n$.
Cons... | Scalar Multiplication Corresponds to Multiplication by 1x1 Matrix | https://proofwiki.org/wiki/Scalar_Multiplication_Corresponds_to_Multiplication_by_1x1_Matrix | https://proofwiki.org/wiki/Scalar_Multiplication_Corresponds_to_Multiplication_by_1x1_Matrix | [
"Matrix Scalar Product",
"Conventional Matrix Multiplication"
] | [
"Definition:Matrix Space",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix Space",
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix Product (Conventional)",
"Definition:Matrix Scalar Product"
] | [
"Definition:Matrix Product (Conventional)",
"Definition:Matrix/Order",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Order",
"Definition:Matrix/Element",
"Definition:Matrix/Indices"
] |
proofwiki-17379 | Closed Unit Ball is Convex Set | Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $\map {B_1^-} 0$ be the closed unit ball in $X$.
Then $\map {B_1^-} 0$ is convex. | Let $x, y \in \map {B_1^-} 0$.
Let $\alpha \in \closedint 0 1$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | l = \norm {\paren {1 - \alpha} x + \alpha y}
| o = \le
| r = \norm {\paren {1 - \alpha} x} + \norm {\alpha y}
| c = {{NormAxiomVector|3}}
}}
{{eqn | r = \size {1 - \alpha} \norm x + \size \alpha \n... | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\map {B_1^-} 0$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $X$.
Then $\map {B_1^-} 0$ is [[Definition:Convex Set (Vector Space)|convex]]. | Let $x, y \in \map {B_1^-} 0$.
Let $\alpha \in \closedint 0 1$ be arbitrary.
Then:
{{begin-eqn}}
{{eqn | l = \norm {\paren {1 - \alpha} x + \alpha y}
| o = \le
| r = \norm {\paren {1 - \alpha} x} + \norm {\alpha y}
| c = {{NormAxiomVector|3}}
}}
{{eqn | r = \size {1 - \alpha} \norm x + \size \alpha... | Closed Unit Ball is Convex Set | https://proofwiki.org/wiki/Closed_Unit_Ball_is_Convex_Set | https://proofwiki.org/wiki/Closed_Unit_Ball_is_Convex_Set | [
"Vector Spaces",
"Closed Balls",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Unit Ball",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Convex Set (Vector Space)"
] |
proofwiki-17380 | Uncountable Sum as Series/Corollary | Let $f: X \to \closedint 0 {+\infty}$ have uncountably infinite support.
Then:
:$\ds \sum_{x \mathop \in X} \map f x = +\infty$ | This is the first case of Uncountable Sum as Series.
{{qed}} | Let $f: X \to \closedint 0 {+\infty}$ have [[Definition:Uncountable Set|uncountably infinite]] [[Definition:Support of Real-Valued Function|support]].
Then:
:$\ds \sum_{x \mathop \in X} \map f x = +\infty$ | This is the first case of [[Uncountable Sum as Series]].
{{qed}} | Uncountable Sum as Series/Corollary | https://proofwiki.org/wiki/Uncountable_Sum_as_Series/Corollary | https://proofwiki.org/wiki/Uncountable_Sum_as_Series/Corollary | [
"Uncountable Sum as Series"
] | [
"Definition:Uncountable/Set",
"Definition:Support of Mapping to Algebraic Structure/Real-Valued Function"
] | [
"Uncountable Sum as Series"
] |
proofwiki-17381 | Identity Matrix from Upper Triangular Matrix | Let $\mathbf A = \sqbrk a_{m n}$ be an upper triangular matrix of order $m \times n$ with no zero diagonal elements.
Let $k = \min \set {m, n}$.
Then $\mathbf A$ can be transformed into a matrix such that the first $k$ rows and columns form the unit matrix of order $k$. | By definition of $k$:
:if $\mathbf A$ has more rows than columns, $k$ is the number of columns of $\mathbf A$.
:if $\mathbf A$ has more columns than rows, $k$ is the number of rows of $\mathbf A$.
Thus let $\mathbf A'$ be the square matrix of order $k$ consisting of the first $k$ rows and columns of $\mathbf A$:
:<nowi... | Let $\mathbf A = \sqbrk a_{m n}$ be an [[Definition:Upper Triangular Matrix|upper triangular matrix]] of [[Definition:Order of Matrix|order]] $m \times n$ with no [[Definition:Zero (Number)|zero]] [[Definition:Diagonal Element|diagonal elements]].
Let $k = \min \set {m, n}$.
Then $\mathbf A$ can be transformed into a... | By definition of $k$:
:if $\mathbf A$ has more [[Definition:Row of Matrix|rows]] than [[Definition:Column of Matrix|columns]], $k$ is the number of [[Definition:Column of Matrix|columns]] of $\mathbf A$.
:if $\mathbf A$ has more [[Definition:Column of Matrix|columns]] than [[Definition:Row of Matrix|rows]], $k$ is the ... | Identity Matrix from Upper Triangular Matrix | https://proofwiki.org/wiki/Identity_Matrix_from_Upper_Triangular_Matrix | https://proofwiki.org/wiki/Identity_Matrix_from_Upper_Triangular_Matrix | [
"Upper Triangular Matrices",
"Identity Matrix from Upper Triangular Matrix"
] | [
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Matrix/Order",
"Definition:Zero (Number)",
"Definition:Main Diagonal/Diagonal Elements",
"Definition:Matrix",
"Definition:Matrix/Row",
"Definition:Matrix/Column",
"Definition:Unit Matrix",
"Definition:Matrix/Square Matrix/Order"
] | [
"Definition:Matrix/Row",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix/Row",
"Definition:Matrix/Column",
"Definit... |
proofwiki-17382 | Simultaneous Linear Equations have Solution iff Ranks of Matrix of Coefficients and Augmented Matrix are Equal | Let $S$ be a system of simultaneous linear equations:
:$\ds \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
Let $S$ be expressed in matrix form as:
:$\mathbf {A x} = \mathbf b$
where:
:$\mathbf A = \begin {pmatrix}
\alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} \\
\alpha_{... | {{ProofWanted|tedious}} | Let $S$ be a [[Definition:Simultaneous Linear Equations|system of simultaneous linear equations]]:
:$\ds \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
Let $S$ be expressed in [[Definition:Matrix Representation of Simultaneous Linear Equations|matrix form]] as:
:$\mathbf {A... | {{ProofWanted|tedious}} | Simultaneous Linear Equations have Solution iff Ranks of Matrix of Coefficients and Augmented Matrix are Equal | https://proofwiki.org/wiki/Simultaneous_Linear_Equations_have_Solution_iff_Ranks_of_Matrix_of_Coefficients_and_Augmented_Matrix_are_Equal | https://proofwiki.org/wiki/Simultaneous_Linear_Equations_have_Solution_iff_Ranks_of_Matrix_of_Coefficients_and_Augmented_Matrix_are_Equal | [
"Simultaneous Linear Equations",
"Rank of Matrix"
] | [
"Definition:Simultaneous Equations/Linear Equations",
"Definition:Simultaneous Linear Equations/Matrix Representation",
"Definition:Simultaneous Equations/Solution",
"Definition:Rank/Matrix",
"Definition:Simultaneous Linear Equations/Matrix Representation/Augmented Matrix"
] | [] |
proofwiki-17383 | Simultaneous Linear Equations has Unique Solution iff Rank of Matrix of Coefficients equals Number of Columns | Let $S$ be a system of $m$ simultaneous linear equations in $n$ variables:
:$\ds \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
Let $S$ be expressed in matrix form as:
:$\mathbf A \mathbf x = \mathbf b$
where:
:$\mathbf A = \begin {pmatrix}
\alpha_{1 1} & \alpha_{1 2} & \cdots... | {{ProofWanted|tedious}} | Let $S$ be a [[Definition:Simultaneous Linear Equations|system of $m$ simultaneous linear equations in $n$ variables]]:
:$\ds \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
Let $S$ be expressed in [[Definition:Matrix Representation of Simultaneous Linear Equations|matrix for... | {{ProofWanted|tedious}} | Simultaneous Linear Equations has Unique Solution iff Rank of Matrix of Coefficients equals Number of Columns | https://proofwiki.org/wiki/Simultaneous_Linear_Equations_has_Unique_Solution_iff_Rank_of_Matrix_of_Coefficients_equals_Number_of_Columns | https://proofwiki.org/wiki/Simultaneous_Linear_Equations_has_Unique_Solution_iff_Rank_of_Matrix_of_Coefficients_equals_Number_of_Columns | [
"Simultaneous Linear Equations",
"Rank of Matrix"
] | [
"Definition:Simultaneous Equations/Linear Equations",
"Definition:Simultaneous Linear Equations/Matrix Representation",
"Definition:Simultaneous Equations/Solution",
"Definition:Rank/Matrix"
] | [] |
proofwiki-17384 | Max Operation Equals an Operand | Let $x_1, x_2, \dotsc, x_n \in S$ for some $n \in \N_{>0}$.
Then:
:$\exists i \in \closedint 1 n : x_i = \max \set {x_1, x_2, \dotsc, x_n}$ | We will prove the result by induction on the number of operands $n$.
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\exists i \in \closedint 1 n : x_i = \max \set {x_1, x_2, \dotsc, x_n}$ | Let $x_1, x_2, \dotsc, x_n \in S$ for some $n \in \N_{>0}$.
Then:
:$\exists i \in \closedint 1 n : x_i = \max \set {x_1, x_2, \dotsc, x_n}$ | We will prove the result by [[Principle of Mathematical Induction|induction]] on the [[Definition:Cardinality|number]] of [[Definition:Operand|operands]] $n$.
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\exists i \in \closedint 1 n : x_i = \max \set {x_1, x_2, \dotsc, x_n}$ | Max Operation Equals an Operand | https://proofwiki.org/wiki/Max_Operation_Equals_an_Operand | https://proofwiki.org/wiki/Max_Operation_Equals_an_Operand | [
"Max Operation"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Cardinality",
"Definition:Operation/Operand",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-17385 | Trace of Sum of Matrices is Sum of Traces | Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be square matrices of order $n$.
let $\mathbf A + \mathbf B$ denote the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
Then:
:$\map \tr {\mathbf A + \mathbf B} = \map \tr {\mathbf A} + \map \tr {\mathbf B}$
where $\map \tr {\mathbf A}$ denotes the trace of... | {{begin-eqn}}
{{eqn | l = \map \tr {\mathbf A} + \map \tr {\mathbf B}
| r = \sum_{k \mathop = 1}^n a_{kk} + \sum_{k \mathop = 1}^n b_{kk}
| c = {{Defof|Trace of Matrix}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \paren {a_{kk} + b_{kk} }
| c = Sum of Summations equals Summation of Sum
}}
{{eqn | r = \map ... | Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order]] $n$.
let $\mathbf A + \mathbf B$ denote the [[Definition:Matrix Entrywise Addition|matrix entrywise sum]] of $\mathbf A$ and $\mathbf B$.
Then:
:$\map \tr {\mathbf A... | {{begin-eqn}}
{{eqn | l = \map \tr {\mathbf A} + \map \tr {\mathbf B}
| r = \sum_{k \mathop = 1}^n a_{kk} + \sum_{k \mathop = 1}^n b_{kk}
| c = {{Defof|Trace of Matrix}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \paren {a_{kk} + b_{kk} }
| c = [[Sum of Summations equals Summation of Sum]]
}}
{{eqn | r = \... | Trace of Sum of Matrices is Sum of Traces | https://proofwiki.org/wiki/Trace_of_Sum_of_Matrices_is_Sum_of_Traces | https://proofwiki.org/wiki/Trace_of_Sum_of_Matrices_is_Sum_of_Traces | [
"Traces of Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix Entrywise Addition",
"Definition:Trace (Linear Algebra)/Matrix"
] | [
"Sum of Summations equals Summation of Sum"
] |
proofwiki-17386 | Similar Matrices have same Traces | Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be square matrices of order $n$.
Let $\mathbf A$ and $\mathbf B$ be similar.
Then:
:$\map \tr {\mathbf A} = \map \tr {\mathbf B}$
where $\map \tr {\mathbf A}$ denotes the trace of $\mathbf A$. | By definition of similar matrices:
:$\exists \mathbf P: \mathbf P^{-1} \mathbf A \mathbf P = \mathbf B$
where $\mathbf P$ is an nonsingular matrix of order $n$.
Therefore:
{{begin-eqn}}
{{eqn | l = \map \tr {\mathbf B}
| r = \map \tr {\mathbf P^{-1} \mathbf A \mathbf P}
}}
{{eqn | r = \map \tr {\mathbf P \paren {... | Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order]] $n$.
Let $\mathbf A$ and $\mathbf B$ be [[Definition:Similar Matrices|similar]].
Then:
:$\map \tr {\mathbf A} = \map \tr {\mathbf B}$
where $\map \tr {\mathbf A}$ d... | By definition of [[Definition:Similar Matrices|similar matrices]]:
:$\exists \mathbf P: \mathbf P^{-1} \mathbf A \mathbf P = \mathbf B$
where $\mathbf P$ is an [[Definition:Nonsingular Matrix|nonsingular matrix]] of [[Definition:Order of Square Matrix|order]] $n$.
Therefore:
{{begin-eqn}}
{{eqn | l = \map \tr {\math... | Similar Matrices have same Traces | https://proofwiki.org/wiki/Similar_Matrices_have_same_Traces | https://proofwiki.org/wiki/Similar_Matrices_have_same_Traces | [
"Matrix Similarity",
"Traces of Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix Similarity",
"Definition:Trace (Linear Algebra)/Matrix"
] | [
"Definition:Matrix Similarity",
"Definition:Nonsingular Matrix",
"Definition:Matrix/Square Matrix/Order",
"Trace of Product of Matrices",
"Matrix Multiplication is Associative",
"Unit Matrix is Identity for Matrix Multiplication"
] |
proofwiki-17387 | Cosine of Integer Multiple of Argument/Formulation 1 | {{begin-eqn}}
{{eqn | l = \cos n \theta
| r = \dfrac 1 2 \paren {\paren {2 \cos \theta}^n - \dfrac n 1 \paren {2 \cos \theta}^{n - 2} + \dfrac n 2 \dbinom {n - 3} 1 \paren {2 \cos \theta}^{n - 4} - \dfrac n 3 \dbinom {n - 4} 2 \paren {2 \cos \theta}^{n - 6} + \cdots}
| c =
}}
{{eqn | r = \dfrac 1 2 \paren ... | The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\ds \cos n \theta = \dfrac 1 2 \paren {\paren {2 \cos \theta}^n + \sum_{k \mathop \ge 1} \paren {-1}^k \dfrac n k \dbinom {n - \paren {k + 1} } {k - 1} \paren {2 \cos \theta}^{n - 2 k} }$ | {{begin-eqn}}
{{eqn | l = \cos n \theta
| r = \dfrac 1 2 \paren {\paren {2 \cos \theta}^n - \dfrac n 1 \paren {2 \cos \theta}^{n - 2} + \dfrac n 2 \dbinom {n - 3} 1 \paren {2 \cos \theta}^{n - 4} - \dfrac n 3 \dbinom {n - 4} 2 \paren {2 \cos \theta}^{n - 6} + \cdots}
| c =
}}
{{eqn | r = \dfrac 1 2 \paren ... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \cos n \theta = \dfrac 1 2 \paren {\paren {2 \cos \theta}^n + \sum_{k \mathop \ge 1} \paren {-1}^k \dfrac n k \dbinom {n - \paren {k + 1} } {k - 1} \par... | Cosine of Integer Multiple of Argument/Formulation 1 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_1 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_1 | [
"Cosine of Integer Multiple of Argument"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-17388 | Cosine of Integer Multiple of Argument/Formulation 2 | {{begin-eqn}}
{{eqn | l = \cos n \theta
| r = \cos^n \theta \paren {1 - \dbinom n 2 \paren {\tan \theta}^2 + \dbinom n 4 \paren {\tan \theta}^4 - \cdots}
| c =
}}
{{eqn | r = \cos^n \theta \sum_{k \mathop \ge 0} \paren {-1}^k \dbinom n {2 k } \paren {\tan^{2 k } \theta}
| c =
}}
{{end-eqn}} | By De Moivre's Formula:
:$\cos n \theta + i \sin n \theta = \paren {\cos \theta + i \sin \theta}^n$
As $n \in \Z_{>0}$, we use the Binomial Theorem on the {{RHS}}, resulting in:
:$\ds \cos n \theta + i \sin n \theta = \sum_{k \mathop \ge 0} \binom n k \paren {\cos^{n - k} \theta} \paren {i \sin \theta}^k$
When $k$ is e... | {{begin-eqn}}
{{eqn | l = \cos n \theta
| r = \cos^n \theta \paren {1 - \dbinom n 2 \paren {\tan \theta}^2 + \dbinom n 4 \paren {\tan \theta}^4 - \cdots}
| c =
}}
{{eqn | r = \cos^n \theta \sum_{k \mathop \ge 0} \paren {-1}^k \dbinom n {2 k } \paren {\tan^{2 k } \theta}
| c =
}}
{{end-eqn}} | By [[De Moivre's Formula]]:
:$\cos n \theta + i \sin n \theta = \paren {\cos \theta + i \sin \theta}^n$
As $n \in \Z_{>0}$, we use the [[Binomial Theorem]] on the {{RHS}}, resulting in:
:$\ds \cos n \theta + i \sin n \theta = \sum_{k \mathop \ge 0} \binom n k \paren {\cos^{n - k} \theta} \paren {i \sin \theta}^k$
Wh... | Cosine of Integer Multiple of Argument/Formulation 2 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_2 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_2 | [
"Cosine of Integer Multiple of Argument"
] | [] | [
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Even Integer",
"Definition:Real Number",
"Definition:Complex Number/Real Part",
"Definition:Even Integer"
] |
proofwiki-17389 | Polygamma Reflection Formula | Let $z \in \C \setminus \Z$.
Let $\psi_n$ denote the $n$th polygamma function.
Then:
:$\map {\psi_n} z - \paren {-1}^n \map {\psi_n} {1 - z} = -\pi \dfrac {\d^n} {\d z^n} \cot \pi z$ | By definition:
:$\map {\psi_n} z = \dfrac {\d^n} {\d z^n} \map \psi z$
where:
:$\psi$ denotes the digamma function
:$z \in \C \setminus \Z_{\le 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \map \psi z - \map \psi {1 - z}
| r = -\pi \cot \pi z
| c = Digamma Reflection Formula
}}
{{eqn | ll= \leadsto
| l = \dfr... | Let $z \in \C \setminus \Z$.
Let $\psi_n$ denote the $n$th [[Definition:Polygamma Function|polygamma function]].
Then:
:$\map {\psi_n} z - \paren {-1}^n \map {\psi_n} {1 - z} = -\pi \dfrac {\d^n} {\d z^n} \cot \pi z$ | By definition:
:$\map {\psi_n} z = \dfrac {\d^n} {\d z^n} \map \psi z$
where:
:$\psi$ denotes the [[Definition:Digamma Function|digamma function]]
:$z \in \C \setminus \Z_{\le 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \map \psi z - \map \psi {1 - z}
| r = -\pi \cot \pi z
| c = [[Digamma Reflection Formula]]
}... | Polygamma Reflection Formula/Proof 1 | https://proofwiki.org/wiki/Polygamma_Reflection_Formula | https://proofwiki.org/wiki/Polygamma_Reflection_Formula/Proof_1 | [
"Polygamma Reflection Formula",
"Polygamma Function",
"Reflection Formulas"
] | [
"Definition:Polygamma Function"
] | [
"Definition:Digamma Function",
"Digamma Reflection Formula",
"Definition:Derivative/Higher Derivatives",
"Definition:Domain (Set Theory)/Mapping"
] |
proofwiki-17390 | Polygamma Reflection Formula | Let $z \in \C \setminus \Z$.
Let $\psi_n$ denote the $n$th polygamma function.
Then:
:$\map {\psi_n} z - \paren {-1}^n \map {\psi_n} {1 - z} = -\pi \dfrac {\d^n} {\d z^n} \cot \pi z$ | {{begin-eqn}}
{{eqn | l = \map \Gamma z \map \Gamma {1 - z}
| r = \dfrac \pi {\sin \pi z}
| c = Euler's Reflection Formula
}}
{{eqn | ll= \leadsto
| l = \map \ln {\map \Gamma z \map \Gamma {1 - z} }
| r = \map \ln {\dfrac \pi {\sin \pi z} }
| c = applying $\ln$ on both sides
}}
{{eqn | ll=... | Let $z \in \C \setminus \Z$.
Let $\psi_n$ denote the $n$th [[Definition:Polygamma Function|polygamma function]].
Then:
:$\map {\psi_n} z - \paren {-1}^n \map {\psi_n} {1 - z} = -\pi \dfrac {\d^n} {\d z^n} \cot \pi z$ | {{begin-eqn}}
{{eqn | l = \map \Gamma z \map \Gamma {1 - z}
| r = \dfrac \pi {\sin \pi z}
| c = [[Euler's Reflection Formula]]
}}
{{eqn | ll= \leadsto
| l = \map \ln {\map \Gamma z \map \Gamma {1 - z} }
| r = \map \ln {\dfrac \pi {\sin \pi z} }
| c = applying $\ln$ on both sides
}}
{{eqn |... | Polygamma Reflection Formula/Proof 2 | https://proofwiki.org/wiki/Polygamma_Reflection_Formula | https://proofwiki.org/wiki/Polygamma_Reflection_Formula/Proof_2 | [
"Polygamma Reflection Formula",
"Polygamma Function",
"Reflection Formulas"
] | [
"Definition:Polygamma Function"
] | [
"Euler's Reflection Formula",
"Sum of Logarithms",
"Difference of Logarithms",
"Definition:Differentiation",
"Derivative of Natural Logarithm Function",
"Derivative of Sine Function",
"Derivative of Composite Function",
"Derivative of Constant",
"Definition:Derivative/Higher Derivatives",
"Definit... |
proofwiki-17391 | Polygamma Function in terms of Hurwitz Zeta Function | :$\map {\psi_n} z = \paren {-1}^{n + 1} \map \Gamma {n + 1} \map \zeta {n + 1, z}$ | {{begin-eqn}}
{{eqn | l = \map \psi z
| r = \dfrac {\map {\Gamma'} z} {\map \Gamma z}
| c = {{Defof|Digamma Function}}
}}
{{eqn | r = -\gamma + \sum_{k \mathop = 1}^\infty \paren {\dfrac 1 k - \dfrac 1 {z + k - 1} }
| c = Reciprocal times Derivative of Gamma Function
}}
{{eqn | ll= \leadsto
| l ... | :$\map {\psi_n} z = \paren {-1}^{n + 1} \map \Gamma {n + 1} \map \zeta {n + 1, z}$ | {{begin-eqn}}
{{eqn | l = \map \psi z
| r = \dfrac {\map {\Gamma'} z} {\map \Gamma z}
| c = {{Defof|Digamma Function}}
}}
{{eqn | r = -\gamma + \sum_{k \mathop = 1}^\infty \paren {\dfrac 1 k - \dfrac 1 {z + k - 1} }
| c = [[Reciprocal times Derivative of Gamma Function]]
}}
{{eqn | ll= \leadsto
... | Polygamma Function in terms of Hurwitz Zeta Function | https://proofwiki.org/wiki/Polygamma_Function_in_terms_of_Hurwitz_Zeta_Function | https://proofwiki.org/wiki/Polygamma_Function_in_terms_of_Hurwitz_Zeta_Function | [
"Hurwitz Zeta Function",
"Polygamma Function"
] | [] | [
"Reciprocal times Derivative of Gamma Function",
"Definition:Derivative",
"Derivative of Constant",
"Nth Derivative of Reciprocal of Mth Power/Corollary",
"Gamma Function Extends Factorial"
] |
proofwiki-17392 | Area of Parallelogram from Determinant | Let $OABC$ be a parallelogram in the Cartesian plane whose vertices are located at:
{{begin-eqn}}
{{eqn | l = O
| r = \tuple {0, 0}
}}
{{eqn | l = A
| r = \tuple {a, c}
}}
{{eqn | l = B
| r = \tuple {a + b, c + d}
}}
{{eqn | l = C
| r = \tuple {b, d}
}}
{{end-eqn}}
The area of $OABC$ is given by... | Arrange for the parallelogram to be situated entirely in the first quadrant.
:500px
First need we establish that $OABC$ is actually a parallelogram in the first place.
Indeed:
{{begin-eqn}}
{{eqn | l = \vec {AB}
| r = \tuple {a + b - a, c + d - c}
| c =
}}
{{eqn | r = \tuple {b, d}
| c =
}}
{{eqn | ... | Let $OABC$ be a [[Definition:Parallelogram|parallelogram]] in the [[Definition:Cartesian Plane|Cartesian plane]] whose [[Definition:Vertex of Polygon|vertices]] are located at:
{{begin-eqn}}
{{eqn | l = O
| r = \tuple {0, 0}
}}
{{eqn | l = A
| r = \tuple {a, c}
}}
{{eqn | l = B
| r = \tuple {a + b, c... | Arrange for the [[Definition:Parallelogram|parallelogram]] to be situated entirely in the [[Definition:First Quadrant|first quadrant]].
:[[File:Area-of-Parallelogram-determinant.png|500px]]
First need we establish that $OABC$ is actually a [[Definition:Parallelogram|parallelogram]] in the first place.
Indeed:
{{b... | Area of Parallelogram from Determinant | https://proofwiki.org/wiki/Area_of_Parallelogram_from_Determinant | https://proofwiki.org/wiki/Area_of_Parallelogram_from_Determinant | [
"Area of Parallelogram"
] | [
"Definition:Quadrilateral/Parallelogram",
"Definition:Cartesian Plane",
"Definition:Polygon/Vertex",
"Definition:Area",
"Determinant/Examples/Order 2"
] | [
"Definition:Quadrilateral/Parallelogram",
"Definition:Cartesian Plane/Quadrants/First",
"File:Area-of-Parallelogram-determinant.png",
"Definition:Quadrilateral/Parallelogram",
"Opposite Sides Equal implies Parallelogram",
"Definition:Quadrilateral/Parallelogram",
"Definition:Area",
"Definition:Area",
... |
proofwiki-17393 | Matrix is Nonsingular iff Rank equals Order | Let $R$ be a commutative ring with unity.
Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$.
Then $\mathbf A$ is nonsingular {{iff}} its rank also equals $n$. | This is an immediate consequence of Square Matrix has Full Rank iff Nonsingular and the definition of full rank.
{{qed}} | Let $R$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $\mathbf A \in R^{n \times n}$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order]] $n$.
Then $\mathbf A$ is [[Definition:Nonsingular Matrix|nonsingular]] {{iff}} its [[Definition:Rank ... | This is an immediate consequence of [[Square Matrix has Full Rank iff Nonsingular]] and the definition of [[Definition:Full Rank|full rank]].
{{qed}} | Matrix is Nonsingular iff Rank equals Order | https://proofwiki.org/wiki/Matrix_is_Nonsingular_iff_Rank_equals_Order | https://proofwiki.org/wiki/Matrix_is_Nonsingular_iff_Rank_equals_Order | [
"Inverse Matrices",
"Rank of Matrix",
"Nonsingular Matrices"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Nonsingular Matrix",
"Definition:Rank/Matrix"
] | [
"Square Matrix has Full Rank iff Nonsingular",
"Definition:Full Rank"
] |
proofwiki-17394 | Determinant of Upper Triangular Matrix | Let $\mathbf T_n$ be an upper triangular matrix of order $n$.
Let $\map \det {\mathbf T_n}$ be the determinant of $\mathbf T_n$.
Then $\map \det {\mathbf T_n}$ is equal to the product of all the diagonal elements of $\mathbf T_n$.
That is:
:$\ds \map \det {\mathbf T_n} = \prod_{k \mathop = 1}^n a_{k k}$ | Let $\mathbf T_n$ be an upper triangular matrix of order $n$.
We proceed by induction on $n$, the number of rows of $\mathbf T_n$. | Let $\mathbf T_n$ be an [[Definition:Upper Triangular Matrix|upper triangular matrix]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\map \det {\mathbf T_n}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf T_n$.
Then $\map \det {\mathbf T_n}$ is equal to the product of all the [[Defini... | Let $\mathbf T_n$ be an [[Definition:Upper Triangular Matrix|upper triangular matrix]] of [[Definition:Order of Square Matrix|order $n$]].
We proceed by [[Principle of Mathematical Induction|induction]] on $n$, the number of [[Definition:Row of Matrix|rows]] of $\mathbf T_n$. | Determinant of Upper Triangular Matrix | https://proofwiki.org/wiki/Determinant_of_Upper_Triangular_Matrix | https://proofwiki.org/wiki/Determinant_of_Upper_Triangular_Matrix | [
"Determinants",
"Upper Triangular Matrices"
] | [
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Main Diagonal/Diagonal Elements"
] | [
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Matrix/Square Matrix/Order",
"Principle of Mathematical Induction",
"Definition:Matrix/Row",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Matrix/Square Matri... |
proofwiki-17395 | Cosine of Integer Multiple of Argument/Formulation 3 | {{begin-eqn}}
{{eqn | l = \cos n \theta
| r = \cos \paren {n - 1} \theta \cos \theta + \paren {1 - \sec^2 \theta} \cos^n \theta \paren {1 + 1 + \frac {\cos 2 \theta} {\cos^2 \theta} + \frac {\cos 3 \theta} {\cos^3 \theta} + \cdots + \frac {\cos \paren {n - 2} \theta} {\cos^{n - 2} \theta} }
| c =
}}
{{eqn ... | {{begin-eqn}}
{{eqn | l = \cos n \theta
| r = \map \cos {\paren {n - 1} \theta + \theta}
| c =
}}
{{eqn | r = \cos \paren {n - 1} \theta \cos \theta - \sin \paren {n - 1} \theta \sin \theta
| c = Cosine of Sum
}}
{{eqn | r = \cos \paren {n - 1} \theta \cos \theta - \paren {\sin \theta \cos^{n - 2} \t... | {{begin-eqn}}
{{eqn | l = \cos n \theta
| r = \cos \paren {n - 1} \theta \cos \theta + \paren {1 - \sec^2 \theta} \cos^n \theta \paren {1 + 1 + \frac {\cos 2 \theta} {\cos^2 \theta} + \frac {\cos 3 \theta} {\cos^3 \theta} + \cdots + \frac {\cos \paren {n - 2} \theta} {\cos^{n - 2} \theta} }
| c =
}}
{{eqn ... | {{begin-eqn}}
{{eqn | l = \cos n \theta
| r = \map \cos {\paren {n - 1} \theta + \theta}
| c =
}}
{{eqn | r = \cos \paren {n - 1} \theta \cos \theta - \sin \paren {n - 1} \theta \sin \theta
| c = [[Cosine of Sum]]
}}
{{eqn | r = \cos \paren {n - 1} \theta \cos \theta - \paren {\sin \theta \cos^{n - 2... | Cosine of Integer Multiple of Argument/Formulation 3 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_3 | https://proofwiki.org/wiki/Cosine_of_Integer_Multiple_of_Argument/Formulation_3 | [
"Cosine of Integer Multiple of Argument"
] | [] | [
"Cosine of Sum",
"Sum of Squares of Sine and Cosine",
"Secant is Reciprocal of Cosine",
"Category:Cosine of Integer Multiple of Argument"
] |
proofwiki-17396 | Determinant of Elementary Row Matrix/Scale Row | Let $e_1$ be the elementary row operation $\text {ERO} 1$:
{{begin-axiom}}
{{axiom | n = \text {ERO} 1
| t = For some $\lambda \ne 0$, multiply row $k$ by $\lambda$
| m = r_k \to \lambda r_k
}}
{{end-axiom}}
which is to operate on some arbitrary matrix space.
Let $\mathbf E_1$ be the elementary row matr... | By Elementary Matrix corresponding to Elementary Row Operation: Scale Row, the elementary row matrix corresponding to $e_1$ is of the form:
:$E_{a b} = \begin {cases} \delta_{a b} & : a \ne k \\ \lambda \cdot \delta_{a b} & : a = k \end{cases}$
where:
:$E_{a b}$ denotes the element of $\mathbf E_1$ whose indices are $\... | Let $e_1$ be the [[Definition:Elementary Row Operation|elementary row operation]] $\text {ERO} 1$:
{{begin-axiom}}
{{axiom | n = \text {ERO} 1
| t = For some $\lambda \ne 0$, [[Definition:Matrix Scalar Product|multiply]] [[Definition:Row of Matrix|row]] $k$ by $\lambda$
| m = r_k \to \lambda r_k
}}
{{e... | By [[Elementary Matrix corresponding to Elementary Row Operation/Scale Row|Elementary Matrix corresponding to Elementary Row Operation: Scale Row]], the [[Definition:Elementary Row Matrix|elementary row matrix]] corresponding to $e_1$ is of the form:
:$E_{a b} = \begin {cases} \delta_{a b} & : a \ne k \\ \lambda \cdot ... | Determinant of Elementary Row Matrix/Scale Row | https://proofwiki.org/wiki/Determinant_of_Elementary_Row_Matrix/Scale_Row | https://proofwiki.org/wiki/Determinant_of_Elementary_Row_Matrix/Scale_Row | [
"Determinant of Elementary Matrix"
] | [
"Definition:Elementary Operation/Row",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Row",
"Definition:Matrix Space",
"Definition:Elementary Matrix/Row Operation",
"Definition:Determinant/Matrix"
] | [
"Elementary Matrix corresponding to Elementary Row Operation/Scale Row",
"Definition:Elementary Matrix/Row Operation",
"Definition:Matrix/Element",
"Definition:Matrix/Indices",
"Definition:Kronecker Delta",
"Definition:Diagonal Matrix",
"Determinant of Diagonal Matrix",
"Definition:Continued Product/I... |
proofwiki-17397 | Determinant of Elementary Row Matrix/Scale Row and Add | Let $e_2$ be the elementary row operation $\text {ERO} 2$:
{{begin-axiom}}
{{axiom | n = \text {ERO} 2
| t = For some $\lambda$, add $\lambda$ times row $j$ to row $i$, where $i \neq j$
| m = r_i \to r_i + \lambda r_j
}}
{{end-axiom}}
which is to operate on some arbitrary matrix space.
Let $\mathbf E_2$... | By Elementary Matrix corresponding to Elementary Row Operation: Scale Row and Add, $\mathbf E_2$ is of the form:
:$E_{a b} = \delta_{a b} + \lambda \cdot \delta_{a i} \cdot \delta_{j b}$
where:
:$E_{a b}$ denotes the element of $\mathbf E$ whose indices are $\tuple {a, b}$
:$\delta_{a b}$ is the Kronecker delta:
::$\de... | Let $e_2$ be the [[Definition:Elementary Row Operation|elementary row operation]] $\text {ERO} 2$:
{{begin-axiom}}
{{axiom | n = \text {ERO} 2
| t = For some $\lambda$, add $\lambda$ [[Definition:Matrix Scalar Product|times]] [[Definition:Row of Matrix|row]] $j$ to [[Definition:Row of Matrix|row]] $i$, where $... | By [[Elementary Matrix corresponding to Elementary Row Operation/Scale Row and Add|Elementary Matrix corresponding to Elementary Row Operation: Scale Row and Add]], $\mathbf E_2$ is of the form:
:$E_{a b} = \delta_{a b} + \lambda \cdot \delta_{a i} \cdot \delta_{j b}$
where:
:$E_{a b}$ denotes the [[Definition:Element... | Determinant of Elementary Row Matrix/Scale Row and Add | https://proofwiki.org/wiki/Determinant_of_Elementary_Row_Matrix/Scale_Row_and_Add | https://proofwiki.org/wiki/Determinant_of_Elementary_Row_Matrix/Scale_Row_and_Add | [
"Determinant of Elementary Matrix"
] | [
"Definition:Elementary Operation/Row",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Matrix Space",
"Definition:Elementary Matrix/Row Operation",
"Definition:Determinant/Matrix"
] | [
"Elementary Matrix corresponding to Elementary Row Operation/Scale Row and Add",
"Definition:Matrix/Element",
"Definition:Matrix/Indices",
"Definition:Kronecker Delta",
"Definition:Main Diagonal/Diagonal Elements",
"Definition:Matrix/Element",
"Definition:Main Diagonal/Diagonal Elements",
"Definition:... |
proofwiki-17398 | Determinant of Elementary Row Matrix/Exchange Rows | Let $e_3$ be the elementary row operation $\text {ERO} 3$:
{{begin-axiom}}
{{axiom | n = \text {ERO} 3
| t = Exchange rows $i$ and $j$
| m = r_i \leftrightarrow r_j
}}
{{end-axiom}}
which is to operate on some arbitrary matrix space.
Let $\mathbf E_3$ be the elementary row matrix corresponding to $e_3$.... | Let $\mathbf I$ denote the unit matrix of arbitrary order $n$.
By Determinant of Unit Matrix:
:$\map \det {\mathbf I} = 1$
Let $\rho$ be the permutation on $\tuple {1, 2, \ldots, n}$ which transposes $i$ and $j$.
From Parity of K-Cycle, $\map \sgn \rho = -1$.
By definition we have that $\mathbf E_3$ is $\mathbf I$ with... | Let $e_3$ be the [[Definition:Elementary Row Operation|elementary row operation]] $\text {ERO} 3$:
{{begin-axiom}}
{{axiom | n = \text {ERO} 3
| t = Exchange [[Definition:Row of Matrix|rows]] $i$ and $j$
| m = r_i \leftrightarrow r_j
}}
{{end-axiom}}
which is to operate on some arbitrary [[Definition:... | Let $\mathbf I$ denote the [[Definition:Unit Matrix|unit matrix]] of arbitrary [[Definition:Order of Square Matrix|order]] $n$.
By [[Determinant of Unit Matrix]]:
:$\map \det {\mathbf I} = 1$
Let $\rho$ be the [[Definition:Permutation on n Letters|permutation]] on $\tuple {1, 2, \ldots, n}$ which [[Definition:Transp... | Determinant of Elementary Row Matrix/Exchange Rows | https://proofwiki.org/wiki/Determinant_of_Elementary_Row_Matrix/Exchange_Rows | https://proofwiki.org/wiki/Determinant_of_Elementary_Row_Matrix/Exchange_Rows | [
"Determinant of Elementary Matrix"
] | [
"Definition:Elementary Operation/Row",
"Definition:Matrix/Row",
"Definition:Matrix Space",
"Definition:Elementary Matrix/Row Operation",
"Definition:Determinant/Matrix"
] | [
"Definition:Unit Matrix",
"Definition:Matrix/Square Matrix/Order",
"Determinant of Unit Matrix",
"Definition:Permutation on n Letters",
"Definition:Transposition",
"Parity of K-Cycle",
"Definition:Matrix/Row",
"Definition:Transposition",
"Definition:Determinant/Matrix",
"Permutation of Determinant... |
proofwiki-17399 | Elementary Matrix corresponding to Elementary Column Operation/Scale Column | Let $e$ be the elementary column operation acting on $\mathbf I$ as:
{{begin-axiom}}
{{axiom | n = \text {ECO} 1
| t = For some $\lambda \in K_{\ne 0}$, multiply column $k$ of $\mathbf I$ by $\lambda$
| m = \kappa_k \to \lambda \kappa_k
}}
{{end-axiom}} | By definition of the unit matrix:
:$I_{a b} = \delta_{a b}$
where:
:$I_{a b}$ denotes the element of $\mathbf I$ whose indices are $\tuple {a, b}$.
By definition, $\mathbf E$ is the square matrix of order $m$ formed by applying $e$ to the unit matrix $\mathbf I$.
That is, all elements of column $k$ of $\mathbf I$ are t... | Let $e$ be the [[Definition:Elementary Column Operation|elementary column operation]] acting on $\mathbf I$ as:
{{begin-axiom}}
{{axiom | n = \text {ECO} 1
| t = For some $\lambda \in K_{\ne 0}$, [[Definition:Matrix Scalar Product|multiply]] [[Definition:Column of Matrix|column]] $k$ of $\mathbf I$ by $\lambda... | By definition of the [[Definition:Unit Matrix|unit matrix]]:
:$I_{a b} = \delta_{a b}$
where:
:$I_{a b}$ denotes the [[Definition:Element of Matrix|element]] of $\mathbf I$ whose [[Definition:Index of Matrix Element|indices]] are $\tuple {a, b}$.
By definition, $\mathbf E$ is the [[Definition:Square Matrix|square mat... | Elementary Matrix corresponding to Elementary Column Operation/Scale Column | https://proofwiki.org/wiki/Elementary_Matrix_corresponding_to_Elementary_Column_Operation/Scale_Column | https://proofwiki.org/wiki/Elementary_Matrix_corresponding_to_Elementary_Column_Operation/Scale_Column | [
"Elementary Matrix corresponding to Elementary Column Operation"
] | [
"Definition:Elementary Operation/Column",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Column"
] | [
"Definition:Unit Matrix",
"Definition:Matrix/Element",
"Definition:Matrix/Indices",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Unit Matrix",
"Definition:Matrix/Element",
"Definition:Matrix/Column",
"Definition:Ring (Abstract Algebra)/Product",
"Definiti... |
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