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-17400 | Elementary Matrix corresponding to Elementary Column Operation/Scale Column and Add | Let $e$ be the elementary column operation acting on $\mathbf I$ as:
{{begin-axiom}}
{{axiom | n = \text {ECO} 2
| t = For some $\lambda \in K$, add $\lambda$ times column $j$ to row $i$
| m = \kappa_i \to \kappa_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 column $i$ 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} 2
| t = For some $\lambda \in K$, add $\lambda$ [[Definition:Matrix Scalar Product|times]] [[Definition:Column of Matrix|column]] $j$ to [[Definition:Row of ... | 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 and Add | https://proofwiki.org/wiki/Elementary_Matrix_corresponding_to_Elementary_Column_Operation/Scale_Column_and_Add | https://proofwiki.org/wiki/Elementary_Matrix_corresponding_to_Elementary_Column_Operation/Scale_Column_and_Add | [
"Elementary Matrix corresponding to Elementary Column Operation"
] | [
"Definition:Elementary Operation/Column",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Column",
"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/Column",
"Definition:Matrix/Element",
"Definition:Matrix/Column"... |
proofwiki-17401 | Elementary Matrix corresponding to Elementary Column Operation/Exchange Columns | Let $e$ be the elementary column operation acting on $\mathbf I$ as:
{{begin-axiom}}
{{axiom | n = \text {ECO} 3
| t = Interchange columns $i$ and $j$
| m = \kappa_i \leftrightarrow \kappa_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 $n$ formed by applying $e$ to the unit matrix $\mathbf I$.
That is, all elements of column $i$ 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} 3
| t = Interchange [[Definition:Column of Matrix|columns]] $i$ and $j$
| m = \kappa_i \leftrightarrow \kappa_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 Column Operation/Exchange Columns | https://proofwiki.org/wiki/Elementary_Matrix_corresponding_to_Elementary_Column_Operation/Exchange_Columns | https://proofwiki.org/wiki/Elementary_Matrix_corresponding_to_Elementary_Column_Operation/Exchange_Columns | [
"Elementary Matrix corresponding to Elementary Column Operation"
] | [
"Definition:Elementary Operation/Column",
"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:Matrix/Element",
"Definition:Matrix/Column"... |
proofwiki-17402 | Column Equivalence is Equivalence Relation | Column equivalence is an equivalence relation. | In the following, $\mathbf A$, $\mathbf B$ and $\mathbf C$ denote arbitrary matrices in a given matrix space $\map \MM {m, n}$ for $m, n \in \Z_{>0}$.
We check in turn each of the conditions for equivalence: | [[Definition:Column Equivalence|Column equivalence]] is an [[Definition:Equivalence Relation|equivalence relation]]. | In the following, $\mathbf A$, $\mathbf B$ and $\mathbf C$ denote arbitrary [[Definition:Matrix|matrices]] in a given [[Definition:Matrix Space|matrix space]] $\map \MM {m, n}$ for $m, n \in \Z_{>0}$.
We check in turn each of the conditions for [[Definition:Equivalence Relation|equivalence]]: | Column Equivalence is Equivalence Relation | https://proofwiki.org/wiki/Column_Equivalence_is_Equivalence_Relation | https://proofwiki.org/wiki/Column_Equivalence_is_Equivalence_Relation | [
"Examples of Equivalence Relations",
"Column Operations"
] | [
"Definition:Column Equivalence",
"Definition:Equivalence Relation"
] | [
"Definition:Matrix",
"Definition:Matrix Space",
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-17403 | Row Operation has Inverse | 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 $\Gamma$ be a row operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$.
Then there exists another row operation $\Gamma'$ which transforms $\mathbf B$ b... | Let $\sequence {e_i}_{1 \mathop \le i \mathop \le k}$ be the finite sequence of elementary row operations that compose $\Gamma$.
Let $\sequence {\mathbf E_i}_{1 \mathop \le i \mathop \le k}$ be the corresponding finite sequence of the elementary row matrices.
From Row Operation is Equivalent to Pre-Multiplication by Pr... | 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 $\Gamma$ be a [[Definition:Row Operation|row operation]] which ... | Let $\sequence {e_i}_{1 \mathop \le i \mathop \le k}$ be the [[Definition:Finite Sequence|finite sequence]] of [[Definition:Elementary Row Operation|elementary row operations]] that compose $\Gamma$.
Let $\sequence {\mathbf E_i}_{1 \mathop \le i \mathop \le k}$ be the corresponding [[Definition:Finite Sequence|finite ... | Row Operation has Inverse | https://proofwiki.org/wiki/Row_Operation_has_Inverse | https://proofwiki.org/wiki/Row_Operation_has_Inverse | [
"Row Operations"
] | [
"Definition:Metric Space",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Matrix",
"Definition:Row Operation",
"Definition:Matrix",
"Definition:Row Operation"
] | [
"Definition:Finite Sequence",
"Definition:Elementary Operation/Row",
"Definition:Finite Sequence",
"Definition:Elementary Matrix/Row Operation",
"Row Operation is Equivalent to Pre-Multiplication by Product of Elementary Matrices",
"Definition:Matrix Product (Conventional)",
"Elementary Row Matrix is No... |
proofwiki-17404 | Real Numbers with Absolute Value form Normed Vector Space | Let $\R$ be the set of real numbers.
Let $\size {\, \cdot \,}$ be the absolute value.
Then $\struct {\R, \size {\, \cdot \,}}$ is a normed vector space. | We have that:
:Real Numbers form Vector Space
:Absolute Value is Norm
By definition, $\struct {\R, \size {\, \cdot \,}}$ is a normed vector space.
{{qed}} | Let $\R$ be the [[Definition:Set|set]] of [[Definition:Real Numbers|real numbers]].
Let $\size {\, \cdot \,}$ be the [[Definition:Absolute Value|absolute value]].
Then $\struct {\R, \size {\, \cdot \,}}$ is a [[Definition:Normed Vector Space|normed vector space]]. | We have that:
:[[Real Numbers form Vector Space]]
:[[Absolute Value is Norm]]
By definition, $\struct {\R, \size {\, \cdot \,}}$ is a [[Definition:Normed Vector Space|normed vector space]].
{{qed}} | Real Numbers with Absolute Value form Normed Vector Space | https://proofwiki.org/wiki/Real_Numbers_with_Absolute_Value_form_Normed_Vector_Space | https://proofwiki.org/wiki/Real_Numbers_with_Absolute_Value_form_Normed_Vector_Space | [
"Examples of Normed Vector Spaces"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Absolute Value",
"Definition:Normed Vector Space"
] | [
"Real Numbers form Vector Space",
"Absolute Value is Norm",
"Definition:Normed Vector Space"
] |
proofwiki-17405 | Finite Dimensional Real Vector Space with Euclidean Norm form Normed Vector Space | Let $\R^n$ be an $n$-dimensional real vector space.
Let $\norm {\, \cdot \,}_2$ be the Euclidean norm.
Then $\struct {\R^n, \norm {\, \cdot \,}_2}$ is a normed vector space. | We have that:
:Real Vector Space is Vector Space
:By Euclidean Space is Normed Vector Space, $\norm {\, \cdot \,}_2$ is a norm on $\R^n$
By definition, $\struct {\R^n, \norm {\, \cdot \,}_2}$ is a normed vector space.
{{qed}} | Let $\R^n$ be an [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Real Vector Space|real vector space]].
Let $\norm {\, \cdot \,}_2$ be the [[Definition:Euclidean Norm|Euclidean norm]].
Then $\struct {\R^n, \norm {\, \cdot \,}_2}$ is a [[Definition:Normed Vector Space|normed vector space]]. | We have that:
:[[Real Vector Space is Vector Space]]
:By [[Euclidean Space is Normed Vector Space]], $\norm {\, \cdot \,}_2$ is a [[Definition:Norm on Vector Space|norm]] on $\R^n$
By definition, $\struct {\R^n, \norm {\, \cdot \,}_2}$ is a [[Definition:Normed Vector Space|normed vector space]].
{{qed}} | Finite Dimensional Real Vector Space with Euclidean Norm form Normed Vector Space | https://proofwiki.org/wiki/Finite_Dimensional_Real_Vector_Space_with_Euclidean_Norm_form_Normed_Vector_Space | https://proofwiki.org/wiki/Finite_Dimensional_Real_Vector_Space_with_Euclidean_Norm_form_Normed_Vector_Space | [
"Examples of Normed Vector Spaces"
] | [
"Definition:Dimension of Vector Space",
"Definition:Real Vector Space",
"Definition:Euclidean Norm",
"Definition:Normed Vector Space"
] | [
"Real Vector Space is Vector Space",
"Euclidean Space is Normed Vector Space",
"Definition:Norm/Vector Space",
"Definition:Normed Vector Space"
] |
proofwiki-17406 | Row Operation is Equivalent to Pre-Multiplication by Product of Elementary Matrices | 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 $\Gamma$ be a row operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$.
Then there exists a unique nonsingular square matrix $\mathbf R$ of order $m$ su... | The proof proceeds by induction.
By definition, $\Gamma$ is a finite sequence of elementary row operations on $\mathbf A$.
Let $\sequence e_k$ denote a finite sequence of elementary row operations $\tuple {e_1, e_2, \ldots, e_k}$ applied on $\mathbf A$ in order: first $e_1$, then $e_2$, then $\ldots$, then $e_k$.
Let $... | 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 $\Gamma$ be a [[Definition:Row Operation|row operation]] which ... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
By definition, $\Gamma$ is a [[Definition:Finite Sequence|finite sequence]] of [[Definition:Elementary Row Operation|elementary row operations]] on $\mathbf A$.
Let $\sequence e_k$ denote a [[Definition:Finite Sequence|finite sequence]] of [[Def... | Row Operation is Equivalent to Pre-Multiplication by Product of Elementary Matrices | https://proofwiki.org/wiki/Row_Operation_is_Equivalent_to_Pre-Multiplication_by_Product_of_Elementary_Matrices | https://proofwiki.org/wiki/Row_Operation_is_Equivalent_to_Pre-Multiplication_by_Product_of_Elementary_Matrices | [
"Row Operations",
"Proofs by Induction"
] | [
"Definition:Metric Space",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Matrix",
"Definition:Row Operation",
"Definition:Matrix",
"Definition:Unique",
"Definition:Nonsingular Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"De... | [
"Principle of Mathematical Induction",
"Definition:Finite Sequence",
"Definition:Elementary Operation/Row",
"Definition:Finite Sequence",
"Definition:Elementary Operation/Row",
"Definition:Row Operation",
"Definition:Elementary Matrix/Row Operation",
"Definition:Matrix/Square Matrix/Order",
"Definit... |
proofwiki-17407 | Product of Matrices is Nonsingular iff Matrices are Nonsingular | Let $\mathbf A$ and $\mathbf B$ be square matrices of order $n$.
Let $\mathbf A \mathbf B$ denote the matrix product of $\mathbf A$ and $\mathbf B$.
Then:
:$\mathbf A \mathbf B$ is nonsingular
{{iff}}
:both $\mathbf A$ and $\mathbf B$ are nonsingular. | === Necessary Condition ===
Let both $\mathbf A$ and $\mathbf B$ be nonsingular.
By Matrix is Nonsingular iff Determinant has Multiplicative Inverse:
:$\map \det {\mathbf A} \ne 0$ and $\map \det {\mathbf B} \ne 0$
where $\map \det {\mathbf A}$ denotes the determinant of $\mathbf A$.
By Determinant of Matrix Product:
:... | Let $\mathbf A$ and $\mathbf B$ be [[Definition:Square Matrix|square matrices of order $n$]].
Let $\mathbf A \mathbf B$ denote the [[Definition:Matrix Product (Conventional)|matrix product]] of $\mathbf A$ and $\mathbf B$.
Then:
:$\mathbf A \mathbf B$ is [[Definition:Nonsingular Matrix|nonsingular]]
{{iff}}
:both $\... | === Necessary Condition ===
Let both $\mathbf A$ and $\mathbf B$ be [[Definition:Nonsingular Matrix|nonsingular]].
By [[Matrix is Nonsingular iff Determinant has Multiplicative Inverse]]:
:$\map \det {\mathbf A} \ne 0$ and $\map \det {\mathbf B} \ne 0$
where $\map \det {\mathbf A}$ denotes the [[Definition:Determinan... | Product of Matrices is Nonsingular iff Matrices are Nonsingular | https://proofwiki.org/wiki/Product_of_Matrices_is_Nonsingular_iff_Matrices_are_Nonsingular | https://proofwiki.org/wiki/Product_of_Matrices_is_Nonsingular_iff_Matrices_are_Nonsingular | [
"Inverse Matrices",
"Conventional Matrix Multiplication"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Nonsingular Matrix",
"Definition:Nonsingular Matrix"
] | [
"Definition:Nonsingular Matrix",
"Matrix is Nonsingular iff Determinant has Multiplicative Inverse",
"Definition:Determinant/Matrix",
"Determinant of Matrix Product",
"Matrix is Nonsingular iff Determinant has Multiplicative Inverse",
"Definition:Nonsingular Matrix",
"Definition:Nonsingular Matrix",
"... |
proofwiki-17408 | Elementary Row Matrix is Nonsingular | Let $\mathbf E$ be an elementary row matrix.
Then $\mathbf E$ is nonsingular. | From Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse it is demonstrated that:
:if $\mathbf E$ is the elementary row matrix corresponding to an elementary row operation $e$
then:
:the inverse of $e$ corresponds to an elementary row matrix which is the inverse of $\mathbf E$.
So as $\mathbf E$ ha... | Let $\mathbf E$ be an [[Definition:Elementary Row Matrix|elementary row matrix]].
Then $\mathbf E$ is [[Definition:Nonsingular Matrix|nonsingular]]. | From [[Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse]] it is demonstrated that:
:if $\mathbf E$ is the [[Definition:Elementary Row Matrix|elementary row matrix]] corresponding to an [[Definition:Elementary Row Operation|elementary row operation]] $e$
then:
:the [[Definition:Inverse of Elemen... | Elementary Row Matrix is Nonsingular | https://proofwiki.org/wiki/Elementary_Row_Matrix_is_Nonsingular | https://proofwiki.org/wiki/Elementary_Row_Matrix_is_Nonsingular | [
"Elementary Matrices",
"Nonsingular Matrices"
] | [
"Definition:Elementary Matrix/Row Operation",
"Definition:Nonsingular Matrix"
] | [
"Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse",
"Definition:Elementary Matrix/Row Operation",
"Definition:Elementary Operation/Row",
"Definition:Inverse of Elementary Row Operation",
"Definition:Elementary Matrix/Row Operation",
"Definition:Inverse Matrix",
"Definition:Invers... |
proofwiki-17409 | Existence of Inverse Elementary Column 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 column 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 $e$.
... | Let us take each type of elementary column 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:
:$\kappa_k$ denote colu... | 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 Column Opera... | Let us take each type of [[Definition:Elementary Column Operation|elementary column 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 demonstrat... | Existence of Inverse Elementary Column Operation | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Column_Operation | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Column_Operation | [
"Elementary Column Operations",
"Existence of Inverse Elementary Column Operation"
] | [
"Definition:Metric Space",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Matrix",
"Definition:Elementary Operation/Column",
"Definition:Matrix",
"Definition:Inverse of Elementary Column Operation",
"Definition:Elementary Operation/Column",
"Definition:Unique"
] | [
"Definition:Elementary Operation/Column",
"Definition:Matrix",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Elementary Operation/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Column... |
proofwiki-17410 | Column Operation has Inverse | 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 $\Gamma$ be a column operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$.
Then there exists another column operation $\Gamma'$ which transforms $\mathb... | Let $\sequence {e_i}_{1 \mathop \le i \mathop \le k}$ be the finite sequence of elementary column operations that compose $\Gamma$.
Let $\sequence {\mathbf E_i}_{1 \mathop \le i \mathop \le k}$ be the corresponding finite sequence of the elementary column matrices.
From Column Operation is Equivalent to Post-Multiplica... | 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 $\Gamma$ be a [[Definition:Column Operation|column operation]] ... | Let $\sequence {e_i}_{1 \mathop \le i \mathop \le k}$ be the [[Definition:Finite Sequence|finite sequence]] of [[Definition:Elementary Column Operation|elementary column operations]] that compose $\Gamma$.
Let $\sequence {\mathbf E_i}_{1 \mathop \le i \mathop \le k}$ be the corresponding [[Definition:Finite Sequence|f... | Column Operation has Inverse | https://proofwiki.org/wiki/Column_Operation_has_Inverse | https://proofwiki.org/wiki/Column_Operation_has_Inverse | [
"Column Operations"
] | [
"Definition:Metric Space",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Matrix",
"Definition:Column Operation",
"Definition:Matrix",
"Definition:Column Operation"
] | [
"Definition:Finite Sequence",
"Definition:Elementary Operation/Column",
"Definition:Finite Sequence",
"Definition:Elementary Matrix/Column Operation",
"Column Operation is Equivalent to Post-Multiplication by Product of Elementary Matrices",
"Definition:Matrix Product (Conventional)",
"Elementary Column... |
proofwiki-17411 | Elementary Column Matrix is Nonsingular | Let $\mathbf E$ be an elementary column matrix.
Then $\mathbf E$ is nonsingular. | From Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse it is demonstrated that:
:if $\mathbf E$ is the elementary column matrix corresponding to an elementary column operation $e$
then:
:the inverse of $e$ corresponds to an elementary column matrix which is the inverse of $\mathbf E$.
So as... | Let $\mathbf E$ be an [[Definition:Elementary Column Matrix|elementary column matrix]].
Then $\mathbf E$ is [[Definition:Nonsingular Matrix|nonsingular]]. | From [[Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse]] it is demonstrated that:
:if $\mathbf E$ is the [[Definition:Elementary Column Matrix|elementary column matrix]] corresponding to an [[Definition:Elementary Column Operation|elementary column operation]] $e$
then:
:the [[Definition... | Elementary Column Matrix is Nonsingular | https://proofwiki.org/wiki/Elementary_Column_Matrix_is_Nonsingular | https://proofwiki.org/wiki/Elementary_Column_Matrix_is_Nonsingular | [
"Elementary Matrices",
"Nonsingular Matrices"
] | [
"Definition:Elementary Matrix/Column Operation",
"Definition:Nonsingular Matrix"
] | [
"Elementary Column Matrix for Inverse of Elementary Column Operation is Inverse",
"Definition:Elementary Matrix/Column Operation",
"Definition:Elementary Operation/Column",
"Definition:Inverse of Elementary Column Operation",
"Definition:Elementary Matrix/Column Operation",
"Definition:Inverse Matrix",
... |
proofwiki-17412 | Column Operation is Equivalent to Post-Multiplication by Product of Elementary Matrices | 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 $\Gamma$ be a column operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$.
Then there exists a unique nonsingular square matrix $\mathbf K$ of order $n$... | The proof proceeds by induction.
By definition, $\Gamma$ is a finite sequence of elementary column operations on $\mathbf A$.
Let $\sequence e_k$ denote a finite sequence of elementary column operations $\tuple {e_1, e_2, \ldots, e_k}$ applied on $\mathbf A$ in order: first $e_1$, then $e_2$, then $\ldots$, then $e_k$.... | 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 $\Gamma$ be a [[Definition:Column Operation|column operation]] ... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
By definition, $\Gamma$ is a [[Definition:Finite Sequence|finite sequence]] of [[Definition:Elementary Column Operation|elementary column operations]] on $\mathbf A$.
Let $\sequence e_k$ denote a [[Definition:Finite Sequence|finite sequence]] of... | Column Operation is Equivalent to Post-Multiplication by Product of Elementary Matrices | https://proofwiki.org/wiki/Column_Operation_is_Equivalent_to_Post-Multiplication_by_Product_of_Elementary_Matrices | https://proofwiki.org/wiki/Column_Operation_is_Equivalent_to_Post-Multiplication_by_Product_of_Elementary_Matrices | [
"Column Operations",
"Proofs by Induction"
] | [
"Definition:Metric Space",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Matrix",
"Definition:Column Operation",
"Definition:Matrix",
"Definition:Unique",
"Definition:Nonsingular Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
... | [
"Principle of Mathematical Induction",
"Definition:Finite Sequence",
"Definition:Elementary Operation/Column",
"Definition:Finite Sequence",
"Definition:Elementary Operation/Column",
"Definition:Column Operation",
"Definition:Elementary Matrix/Column Operation",
"Definition:Matrix/Square Matrix/Order"... |
proofwiki-17413 | Existence of Inverse Elementary Row Operation/Scalar Product of Row | 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 the elementary row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$.
{{begin-axiom}}
{{axiom | n = \text {ERO} 1
| ... | Let $\map e {\mathbf A}$ be the elementary row operation:
:$e := r_k \to \lambda r_k$
where $\lambda \ne 0$.
Then $r'_k$ is such that:
:$\forall a'_{k i} \in r'_k: a'_{k i} = \lambda a_{k i}$
Now let $\map {e'} {\mathbf A'}$ be the elementary row operation which transforms $\mathbf A'$ to $\mathbf A' '$:
:$e' := r_k \t... | 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 the [[Definition:Elementary Row Operat... | Let $\map e {\mathbf A}$ be the [[Definition:Elementary Row Operation|elementary row operation]]:
:$e := r_k \to \lambda r_k$
where $\lambda \ne 0$.
Then $r'_k$ is such that:
:$\forall a'_{k i} \in r'_k: a'_{k i} = \lambda a_{k i}$
Now let $\map {e'} {\mathbf A'}$ be the [[Definition:Elementary Row Operation|elemen... | Existence of Inverse Elementary Row Operation/Scalar Product of Row | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Row_Operation/Scalar_Product_of_Row | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Row_Operation/Scalar_Product_of_Row | [
"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:Matrix Scalar Product",
"Definition:Matrix/Row",
"Definition:Inverse of Elementary Row Operation",
"Definition:E... | [
"Definition:Elementary Operation/Row",
"Definition:Elementary Operation/Row",
"Definition:Elementary Operation/Row",
"Definition:Field (Abstract Algebra)",
"Definition:Elementary Operation/Row"
] |
proofwiki-17414 | Existence of Inverse Elementary Row Operation/Add Scalar Product of Row to Another | 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 the elementary row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$.
{{begin-axiom}}
{{axiom | n = \text {ERO} 2
| ... | Let $\map e {\mathbf A}$ be the elementary row operation:
:$e := r_k \to r_k + \lambda r_l$
Then $r'_k$ is such that:
:$\forall a'_{k i} \in r'_k: a'_{k i} = a_{k i} + \lambda a_{l i}$
Now let $\map {e'} {\mathbf A'}$ be the elementary row operation which transforms $\mathbf A'$ to $\mathbf A' '$:
:$e' := r'_k \to r'_k... | 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 the [[Definition:Elementary Row Operati... | Let $\map e {\mathbf A}$ be the [[Definition:Elementary Row Operation|elementary row operation]]:
:$e := r_k \to r_k + \lambda r_l$
Then $r'_k$ is such that:
:$\forall a'_{k i} \in r'_k: a'_{k i} = a_{k i} + \lambda a_{l i}$
Now let $\map {e'} {\mathbf A'}$ be the [[Definition:Elementary Row Operation|elementary ro... | Existence of Inverse Elementary Row Operation/Add Scalar Product of Row to Another | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Row_Operation/Add_Scalar_Product_of_Row_to_Another | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Row_Operation/Add_Scalar_Product_of_Row_to_Another | [
"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:Matrix Scalar Product",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Inverse of Elementary Row ... | [
"Definition:Elementary Operation/Row",
"Definition:Elementary Operation/Row",
"Definition:Matrix/Row",
"Definition:Elementary Operation/Row"
] |
proofwiki-17415 | Existence of Inverse Elementary Row Operation/Exchange Rows | 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 the elementary row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$.
{{begin-axiom}}
{{axiom | n = \text {ERO} 3
| ... | Let $\map e {\mathbf A}$ be the elementary row operation:
:$e := r_k \leftrightarrow r_l$
Thus we have:
{{begin-eqn}}
{{eqn | l = r'_k
| r = r_l
| c =
}}
{{eqn | lo= \text {and}
| l = r'_l
| r = r_k
| c =
}}
{{end-eqn}}
Now let $\map {e'} {\mathbf A'}$ be the elementary row operation whi... | 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 the [[Definition:Elementary Row Operat... | Let $\map e {\mathbf A}$ be the [[Definition:Elementary Row Operation|elementary row operation]]:
:$e := r_k \leftrightarrow r_l$
Thus we have:
{{begin-eqn}}
{{eqn | l = r'_k
| r = r_l
| c =
}}
{{eqn | lo= \text {and}
| l = r'_l
| r = r_k
| c =
}}
{{end-eqn}}
Now let $\map {e'} {\ma... | Existence of Inverse Elementary Row Operation/Exchange Rows | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Row_Operation/Exchange_Rows | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Row_Operation/Exchange_Rows | [
"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:Matrix/Row",
"Definition:Inverse of Elementary Row Operation",
"Definition:Elementary Operation/Row"
] | [
"Definition:Elementary Operation/Row",
"Definition:Elementary Operation/Row",
"Definition:Elementary Operation/Row"
] |
proofwiki-17416 | Existence of Inverse Elementary Column Operation/Scalar Product of Column | 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 the elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$.
{{begin-axiom}}
{{axiom | n = \text {ECO} 1
... | Let $\map e {\mathbf A}$ be the elementary column operation:
:$e := \kappa_k \to \lambda \kappa_k$
where $\lambda \ne 0$.
Then $\kappa'_k$ is such that:
:$\forall a'_{k i} \in \kappa'_k: a'_{k i} = \lambda a_{k i}$
Now let $\map {e'} {\mathbf A'}$ be the elementary column operation which transforms $\mathbf A'$ to $\ma... | 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 the [[Definition:Elementary Column Ope... | Let $\map e {\mathbf A}$ be the [[Definition:Elementary Column Operation|elementary column operation]]:
:$e := \kappa_k \to \lambda \kappa_k$
where $\lambda \ne 0$.
Then $\kappa'_k$ is such that:
:$\forall a'_{k i} \in \kappa'_k: a'_{k i} = \lambda a_{k i}$
Now let $\map {e'} {\mathbf A'}$ be the [[Definition:Eleme... | Existence of Inverse Elementary Column Operation/Scalar Product of Column | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Column_Operation/Scalar_Product_of_Column | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Column_Operation/Scalar_Product_of_Column | [
"Existence of Inverse Elementary Column Operation"
] | [
"Definition:Metric Space",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Matrix",
"Definition:Elementary Operation/Column",
"Definition:Matrix",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Column",
"Definition:Inverse of Elementary Column Operation",
"Def... | [
"Definition:Elementary Operation/Column",
"Definition:Elementary Operation/Column",
"Definition:Elementary Operation/Column",
"Definition:Field (Abstract Algebra)",
"Definition:Elementary Operation/Column"
] |
proofwiki-17417 | Existence of Inverse Elementary Column Operation/Add Scalar Product of Column to Another | 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 the elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$.
{{begin-axiom}}
{{axiom | n = \text {ECO} 2
... | Let $\map e {\mathbf A}$ be the elementary column operation:
:$e := \kappa_k \to \kappa_k + \lambda r_l$
Then $\kappa'_k$ is such that:
:$\forall a'_{i k} \in \kappa'_k: a'_{i k} = a_{i k} + \lambda a_{i l}$
Now let $\map {e'} {\mathbf A'}$ be the elementary column operation which transforms $\mathbf A'$ to $\mathbf A'... | 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 the [[Definition:Elementary Column Ope... | Let $\map e {\mathbf A}$ be the [[Definition:Elementary Column Operation|elementary column operation]]:
:$e := \kappa_k \to \kappa_k + \lambda r_l$
Then $\kappa'_k$ is such that:
:$\forall a'_{i k} \in \kappa'_k: a'_{i k} = a_{i k} + \lambda a_{i l}$
Now let $\map {e'} {\mathbf A'}$ be the [[Definition:Elementary C... | Existence of Inverse Elementary Column Operation/Add Scalar Product of Column to Another | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Column_Operation/Add_Scalar_Product_of_Column_to_Another | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Column_Operation/Add_Scalar_Product_of_Column_to_Another | [
"Existence of Inverse Elementary Column Operation"
] | [
"Definition:Metric Space",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Matrix",
"Definition:Elementary Operation/Column",
"Definition:Matrix",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Inverse of Elemen... | [
"Definition:Elementary Operation/Column",
"Definition:Elementary Operation/Column",
"Definition:Matrix/Column",
"Definition:Elementary Operation/Column"
] |
proofwiki-17418 | Existence of Inverse Elementary Column Operation/Exchange Columns | 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 the elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$.
{{begin-axiom}}
{{axiom | n = \text {ECO} 3
... | Let $\map e {\mathbf A}$ be the elementary column operation:
:$e := \kappa_k \leftrightarrow \kappa_l$
Thus we have:
{{begin-eqn}}
{{eqn | l = \kappa'_k
| r = \kappa_l
| c =
}}
{{eqn | lo= \text {and}
| l = \kappa'_l
| r = \kappa_k
| c =
}}
{{end-eqn}}
Now let $\map {e'} {\mathbf A'}$ be... | 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 the [[Definition:Elementary Column Ope... | Let $\map e {\mathbf A}$ be the [[Definition:Elementary Column Operation|elementary column operation]]:
:$e := \kappa_k \leftrightarrow \kappa_l$
Thus we have:
{{begin-eqn}}
{{eqn | l = \kappa'_k
| r = \kappa_l
| c =
}}
{{eqn | lo= \text {and}
| l = \kappa'_l
| r = \kappa_k
| c =
}}
{... | Existence of Inverse Elementary Column Operation/Exchange Columns | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Column_Operation/Exchange_Columns | https://proofwiki.org/wiki/Existence_of_Inverse_Elementary_Column_Operation/Exchange_Columns | [
"Existence of Inverse Elementary Column Operation"
] | [
"Definition:Metric Space",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Matrix",
"Definition:Elementary Operation/Column",
"Definition:Matrix",
"Definition:Matrix/Column",
"Definition:Inverse of Elementary Column Operation",
"Definition:Elementary Operation/Column"
] | [
"Definition:Elementary Operation/Column",
"Definition:Elementary Operation/Column",
"Definition:Elementary Operation/Column"
] |
proofwiki-17419 | Elementary Column Operations as Matrix Multiplications | Let $e$ be an elementary column operation.
Let $\mathbf E$ be the elementary column matrix of order $n$ defined as:
:$\mathbf E = e \paren {\mathbf I}$
where $\mathbf I$ is the unit matrix.
Then for every $m \times n$ matrix $\mathbf A$:
:$e \paren {\mathbf A} = \mathbf A \mathbf E$
where $\mathbf A \mathbf E$ denotes ... | Let $s, t \in \closedint 1 m$ such that $s \ne t$. | Let $e$ be an [[Definition:Elementary Column Operation|elementary column operation]].
Let $\mathbf E$ be the [[Definition:Elementary Column Matrix|elementary column matrix]] of [[Definition:Order of Square Matrix|order]] $n$ defined as:
:$\mathbf E = e \paren {\mathbf I}$
where $\mathbf I$ is the [[Definition:Unit Mat... | Let $s, t \in \closedint 1 m$ such that $s \ne t$. | Elementary Column Operations as Matrix Multiplications | https://proofwiki.org/wiki/Elementary_Column_Operations_as_Matrix_Multiplications | https://proofwiki.org/wiki/Elementary_Column_Operations_as_Matrix_Multiplications | [
"Conventional Matrix Multiplication",
"Elementary Column Operations",
"Elementary Matrices"
] | [
"Definition:Elementary Operation/Column",
"Definition:Elementary Matrix/Column Operation",
"Definition:Matrix/Square Matrix/Order",
"Definition:Unit Matrix",
"Definition:Matrix",
"Definition:Matrix Product (Conventional)"
] | [] |
proofwiki-17420 | Square Root of Number Plus Square Root | :$\ds \sqrt {a + \sqrt b} = \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} + \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}$ | We are given that $a^2 - b > 0$.
Then:
:$a > \sqrt b$
and so $\ds \sqrt {a + \sqrt b}$ is defined on the real numbers.
Let $\ds \sqrt {a + \sqrt b} = \sqrt x + \sqrt y$ where $x, y$ are (strictly) positive real numbers.
Squaring both sides gives:
{{begin-eqn}}
{{eqn | l = a + \sqrt b
| r = \paren {\sqrt x + \sqrt... | :$\ds \sqrt {a + \sqrt b} = \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} + \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}$ | We are given that $a^2 - b > 0$.
Then:
:$a > \sqrt b$
and so $\ds \sqrt {a + \sqrt b}$ is defined on the [[Definition:Real Number|real numbers]].
Let $\ds \sqrt {a + \sqrt b} = \sqrt x + \sqrt y$ where $x, y$ are [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]].
[[Definition:Square Func... | Square Root of Number Plus Square Root/Proof 1 | https://proofwiki.org/wiki/Square_Root_of_Number_Plus_Square_Root | https://proofwiki.org/wiki/Square_Root_of_Number_Plus_Square_Root/Proof_1 | [
"Square Root of Number Plus or Minus Square Root"
] | [] | [
"Definition:Real Number",
"Definition:Strictly Positive/Real Number",
"Definition:Square/Function",
"Cardano's Formula",
"Viète's Formulas",
"Definition:Quadratic Equation",
"Solution to Quadratic Equation"
] |
proofwiki-17421 | Square Root of Number Plus Square Root | :$\ds \sqrt {a + \sqrt b} = \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} + \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}$ | {{begin-eqn}}
{{eqn | l = \paren {\sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} + \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2} }^2
| r = \dfrac {a + \sqrt {a^2 - b} } 2 + \dfrac {a - \sqrt {a^2 - b} } 2 + 2 \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}
| c = multiplying out
}}
{{eqn |... | :$\ds \sqrt {a + \sqrt b} = \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} + \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}$ | {{begin-eqn}}
{{eqn | l = \paren {\sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} + \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2} }^2
| r = \dfrac {a + \sqrt {a^2 - b} } 2 + \dfrac {a - \sqrt {a^2 - b} } 2 + 2 \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}
| c = multiplying out
}}
{{eqn |... | Square Root of Number Plus Square Root/Proof 2 | https://proofwiki.org/wiki/Square_Root_of_Number_Plus_Square_Root | https://proofwiki.org/wiki/Square_Root_of_Number_Plus_Square_Root/Proof_2 | [
"Square Root of Number Plus or Minus Square Root"
] | [] | [
"Difference of Two Squares",
"Definition:Square Root"
] |
proofwiki-17422 | Square Root of Number Plus Square Root/Proof 1 | Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$.
Then:
{{:Square Root of Number Plus Square Root}} | We are given that $a^2 - b > 0$.
Then:
:$a > \sqrt b$
and so $\ds \sqrt {a + \sqrt b}$ is defined on the real numbers.
Let $\ds \sqrt {a + \sqrt b} = \sqrt x + \sqrt y$ where $x, y$ are (strictly) positive real numbers.
Squaring both sides gives:
{{begin-eqn}}
{{eqn | l = a + \sqrt b
| r = \paren {\sqrt x + \sqrt... | Let $a$ and $b$ be [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]] such that $a^2 - b > 0$.
Then:
{{:Square Root of Number Plus Square Root}} | We are given that $a^2 - b > 0$.
Then:
:$a > \sqrt b$
and so $\ds \sqrt {a + \sqrt b}$ is defined on the [[Definition:Real Number|real numbers]].
Let $\ds \sqrt {a + \sqrt b} = \sqrt x + \sqrt y$ where $x, y$ are [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]].
[[Definition:Square Func... | Square Root of Number Plus Square Root/Proof 1 | https://proofwiki.org/wiki/Square_Root_of_Number_Plus_Square_Root/Proof_1 | https://proofwiki.org/wiki/Square_Root_of_Number_Plus_Square_Root/Proof_1 | [
"Square Root of Number Plus or Minus Square Root"
] | [
"Definition:Strictly Positive/Real Number"
] | [
"Definition:Real Number",
"Definition:Strictly Positive/Real Number",
"Definition:Square/Function",
"Cardano's Formula",
"Viète's Formulas",
"Definition:Quadratic Equation",
"Solution to Quadratic Equation"
] |
proofwiki-17423 | Square Root of Number Plus Square Root/Proof 2 | Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$.
Then:
{{:Square Root of Number Plus Square Root}} | {{begin-eqn}}
{{eqn | l = \paren {\sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} + \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2} }^2
| r = \dfrac {a + \sqrt {a^2 - b} } 2 + \dfrac {a - \sqrt {a^2 - b} } 2 + 2 \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}
| c = multiplying out
}}
{{eqn |... | Let $a$ and $b$ be [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]] such that $a^2 - b > 0$.
Then:
{{:Square Root of Number Plus Square Root}} | {{begin-eqn}}
{{eqn | l = \paren {\sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} + \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2} }^2
| r = \dfrac {a + \sqrt {a^2 - b} } 2 + \dfrac {a - \sqrt {a^2 - b} } 2 + 2 \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}
| c = multiplying out
}}
{{eqn |... | Square Root of Number Plus Square Root/Proof 2 | https://proofwiki.org/wiki/Square_Root_of_Number_Plus_Square_Root/Proof_2 | https://proofwiki.org/wiki/Square_Root_of_Number_Plus_Square_Root/Proof_2 | [
"Square Root of Number Plus or Minus Square Root"
] | [
"Definition:Strictly Positive/Real Number"
] | [
"Difference of Two Squares",
"Definition:Square Root"
] |
proofwiki-17424 | Square Root of Number Minus Square Root | :$\ds \sqrt {a - \sqrt b} = \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} - \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}$ | We are given that $a^2 - b > 0$.
Then:
:$a > \sqrt b$
and so $\ds \sqrt {a - \sqrt b}$ is defined on the real numbers.
Let $\ds \sqrt {a - \sqrt b} = \sqrt x - \sqrt y$ where $x, y$ are (strictly) positive real numbers.
Observe that:
:$\ds 0 < \sqrt {a - \sqrt b} = \sqrt x - \sqrt y \implies x > y$
Squaring both sides ... | :$\ds \sqrt {a - \sqrt b} = \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} - \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}$ | We are given that $a^2 - b > 0$.
Then:
:$a > \sqrt b$
and so $\ds \sqrt {a - \sqrt b}$ is defined on the [[Definition:Real Number|real numbers]].
Let $\ds \sqrt {a - \sqrt b} = \sqrt x - \sqrt y$ where $x, y$ are [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]].
Observe that:
:$\ds 0 < ... | Square Root of Number Minus Square Root/Proof 1 | https://proofwiki.org/wiki/Square_Root_of_Number_Minus_Square_Root | https://proofwiki.org/wiki/Square_Root_of_Number_Minus_Square_Root/Proof_1 | [
"Square Root of Number Plus or Minus Square Root"
] | [] | [
"Definition:Real Number",
"Definition:Strictly Positive/Real Number",
"Definition:Square/Function",
"Viète's Formulas",
"Definition:Quadratic Equation",
"Solution to Quadratic Equation"
] |
proofwiki-17425 | Square Root of Number Minus Square Root | :$\ds \sqrt {a - \sqrt b} = \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} - \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}$ | {{begin-eqn}}
{{eqn | l = \paren {\sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} - \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2} }^2
| r = \dfrac {a + \sqrt {a^2 - b} } 2 + \dfrac {a - \sqrt {a^2 - b} } 2 - 2 \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}
| c = multiplying out
}}
{{eqn |... | :$\ds \sqrt {a - \sqrt b} = \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} - \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}$ | {{begin-eqn}}
{{eqn | l = \paren {\sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} - \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2} }^2
| r = \dfrac {a + \sqrt {a^2 - b} } 2 + \dfrac {a - \sqrt {a^2 - b} } 2 - 2 \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}
| c = multiplying out
}}
{{eqn |... | Square Root of Number Minus Square Root/Proof 2 | https://proofwiki.org/wiki/Square_Root_of_Number_Minus_Square_Root | https://proofwiki.org/wiki/Square_Root_of_Number_Minus_Square_Root/Proof_2 | [
"Square Root of Number Plus or Minus Square Root"
] | [] | [
"Difference of Two Squares",
"Definition:Square Root"
] |
proofwiki-17426 | Square Root of Number Minus Square Root/Proof 2 | Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$.
Then:
{{:Square Root of Number Minus Square Root}} | {{begin-eqn}}
{{eqn | l = \paren {\sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} - \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2} }^2
| r = \dfrac {a + \sqrt {a^2 - b} } 2 + \dfrac {a - \sqrt {a^2 - b} } 2 - 2 \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}
| c = multiplying out
}}
{{eqn |... | Let $a$ and $b$ be [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]] such that $a^2 - b > 0$.
Then:
{{:Square Root of Number Minus Square Root}} | {{begin-eqn}}
{{eqn | l = \paren {\sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} - \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2} }^2
| r = \dfrac {a + \sqrt {a^2 - b} } 2 + \dfrac {a - \sqrt {a^2 - b} } 2 - 2 \sqrt {\dfrac {a + \sqrt {a^2 - b} } 2} \sqrt {\dfrac {a - \sqrt {a^2 - b} } 2}
| c = multiplying out
}}
{{eqn |... | Square Root of Number Minus Square Root/Proof 2 | https://proofwiki.org/wiki/Square_Root_of_Number_Minus_Square_Root/Proof_2 | https://proofwiki.org/wiki/Square_Root_of_Number_Minus_Square_Root/Proof_2 | [
"Square Root of Number Plus or Minus Square Root"
] | [
"Definition:Strictly Positive/Real Number"
] | [
"Difference of Two Squares",
"Definition:Square Root"
] |
proofwiki-17427 | Square Root of Number Minus Square Root/Proof 1 | Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$.
Then:
{{:Square Root of Number Minus Square Root}} | We are given that $a^2 - b > 0$.
Then:
:$a > \sqrt b$
and so $\ds \sqrt {a - \sqrt b}$ is defined on the real numbers.
Let $\ds \sqrt {a - \sqrt b} = \sqrt x - \sqrt y$ where $x, y$ are (strictly) positive real numbers.
Observe that:
:$\ds 0 < \sqrt {a - \sqrt b} = \sqrt x - \sqrt y \implies x > y$
Squaring both sides ... | Let $a$ and $b$ be [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]] such that $a^2 - b > 0$.
Then:
{{:Square Root of Number Minus Square Root}} | We are given that $a^2 - b > 0$.
Then:
:$a > \sqrt b$
and so $\ds \sqrt {a - \sqrt b}$ is defined on the [[Definition:Real Number|real numbers]].
Let $\ds \sqrt {a - \sqrt b} = \sqrt x - \sqrt y$ where $x, y$ are [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]].
Observe that:
:$\ds 0 < ... | Square Root of Number Minus Square Root/Proof 1 | https://proofwiki.org/wiki/Square_Root_of_Number_Minus_Square_Root/Proof_1 | https://proofwiki.org/wiki/Square_Root_of_Number_Minus_Square_Root/Proof_1 | [
"Square Root of Number Plus or Minus Square Root"
] | [
"Definition:Strictly Positive/Real Number"
] | [
"Definition:Real Number",
"Definition:Strictly Positive/Real Number",
"Definition:Square/Function",
"Viète's Formulas",
"Definition:Quadratic Equation",
"Solution to Quadratic Equation"
] |
proofwiki-17428 | Exchange of Columns as Sequence of Other Elementary Column Operations | Let $\mathbf A$ be an $m \times n$ matrix.
Let $i, j \in \closedint 1 m: i \ne j$
Let $\kappa_k$ denote the $k$th column of $\mathbf A$ for $1 \le k \le n$:
:$\kappa_k = \begin {pmatrix} a_{1 k} \\ a_{2 k} \\ \vdots \\ a_{m k} \end {pmatrix}$
Let $e$ be the elementary column operation acting on $\mathbf A$ as:
{{begin-... | In the below:
:$\kappa_i$ denotes the initial state of column $i$
:$\kappa_j$ denotes the initial state of column $j$
:$\kappa_i'$ denotes the state of column $i$ after having had the latest elementary column operation applied
:$\kappa_j'$ denotes the state of column $j$ after having had the latest elementary column op... | Let $\mathbf A$ be an $m \times n$ [[Definition:Matrix|matrix]].
Let $i, j \in \closedint 1 m: i \ne j$
Let $\kappa_k$ denote the $k$th [[Definition:Column of Matrix|column]] of $\mathbf A$ for $1 \le k \le n$:
:$\kappa_k = \begin {pmatrix} a_{1 k} \\ a_{2 k} \\ \vdots \\ a_{m k} \end {pmatrix}$
Let $e$ be the [[De... | In the below:
:$\kappa_i$ denotes the initial state of [[Definition:Column of Matrix|column]] $i$
:$\kappa_j$ denotes the initial state of [[Definition:Column of Matrix|column]] $j$
:$\kappa_i'$ denotes the state of [[Definition:Column of Matrix|column]] $i$ after having had the latest [[Definition:Elementary Column O... | Exchange of Columns as Sequence of Other Elementary Column Operations | https://proofwiki.org/wiki/Exchange_of_Columns_as_Sequence_of_Other_Elementary_Column_Operations | https://proofwiki.org/wiki/Exchange_of_Columns_as_Sequence_of_Other_Elementary_Column_Operations | [
"Elementary Column Operations"
] | [
"Definition:Matrix",
"Definition:Matrix/Column",
"Definition:Elementary Operation/Column",
"Definition:Matrix/Column",
"Definition:Finite Sequence",
"Definition:Elementary Operation/Column",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Column",
"Definition:Matrix Scalar Product",
"Defini... | [
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Elementary Operation/Column",
"Definition:Matrix/Column",
"Definition:Elementary Operation/Column",
"Definition:Elementary Operation/Column",
"Definition:Matrix/Column",
"Definition:Elementary Operation/C... |
proofwiki-17429 | Effect of Elementary Column Operations on Determinant | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $\map \det {\mathbf A}$ denote the determinant of $\mathbf A$.
Take the elementary column operations:
{{begin-axiom}}
{{axiom | n = \text {ECO} 1
| t = For some $\lambda$, multiply column $i$ by $\lambda$
| m = \kappa_i \to \lambda \kappa... | From Elementary Column Operations as Matrix Multiplications, an elementary column operation on $\mathbf A$ is equivalent to matrix multiplication by the elementary column matrices corresponding to the elementary column operations.
From Determinant of Elementary Column Matrix, the determinants of those elementary column... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix of order $n$]].
Let $\map \det {\mathbf A}$ denote the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Take the [[Definition:Elementary Column Operation|elementary column operations]]:
{{begin-axiom}}
{{axiom | n = \text {EC... | From [[Elementary Column Operations as Matrix Multiplications]], an [[Definition:Elementary Column Operation|elementary column operation]] on $\mathbf A$ is equivalent to [[Definition:Matrix Product (Conventional)|matrix multiplication]] by the [[Definition:Elementary Column Matrix|elementary column matrices]] correspo... | Effect of Elementary Column Operations on Determinant | https://proofwiki.org/wiki/Effect_of_Elementary_Column_Operations_on_Determinant | https://proofwiki.org/wiki/Effect_of_Elementary_Column_Operations_on_Determinant | [
"Determinants",
"Elementary Column Operations"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Determinant/Matrix",
"Definition:Elementary Operation/Column",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Column",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Column"
] | [
"Elementary Column Operations as Matrix Multiplications",
"Definition:Elementary Operation/Column",
"Definition:Matrix Product (Conventional)",
"Definition:Elementary Matrix/Column Operation",
"Definition:Elementary Operation/Column",
"Determinant of Elementary Column Matrix",
"Definition:Determinant/Ma... |
proofwiki-17430 | Determinant of Elementary Column Matrix/Scale Column | Let $e_1$ be the elementary column operation $\text {ECO} 1$:
{{begin-axiom}}
{{axiom | n = \text {ECO} 1
| t = For some $\lambda \ne 0$, multiply column $k$ by $\lambda$
| m = \kappa_k \to \lambda \kappa_k
}}
{{end-axiom}}
which is to operate on some arbitrary matrix space.
Let $\mathbf E_1$ be the ele... | By Elementary Matrix corresponding to Elementary Column Operation: Scale Column, the elementary column 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 indic... | Let $e_1$ be the [[Definition:Elementary Column Operation|elementary column operation]] $\text {ECO} 1$:
{{begin-axiom}}
{{axiom | n = \text {ECO} 1
| t = For some $\lambda \ne 0$, [[Definition:Matrix Scalar Product|multiply]] [[Definition:Column of Matrix|column]] $k$ by $\lambda$
| m = \kappa_k \to \... | By [[Elementary Matrix corresponding to Elementary Column Operation/Scale Column|Elementary Matrix corresponding to Elementary Column Operation: Scale Column]], the [[Definition:Elementary Column Matrix|elementary column matrix]] corresponding to $e_1$ is of the form:
:$E_{a b} = \begin {cases} \delta_{a b} & : a \ne k... | Determinant of Elementary Column Matrix/Scale Column | https://proofwiki.org/wiki/Determinant_of_Elementary_Column_Matrix/Scale_Column | https://proofwiki.org/wiki/Determinant_of_Elementary_Column_Matrix/Scale_Column | [
"Determinant of Elementary Matrix"
] | [
"Definition:Elementary Operation/Column",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Column",
"Definition:Matrix Space",
"Definition:Elementary Matrix/Column Operation",
"Definition:Determinant/Matrix"
] | [
"Elementary Matrix corresponding to Elementary Column Operation/Scale Column",
"Definition:Elementary Matrix/Column Operation",
"Definition:Matrix/Element",
"Definition:Matrix/Indices",
"Definition:Kronecker Delta",
"Definition:Diagonal Matrix",
"Determinant of Diagonal Matrix",
"Definition:Continued ... |
proofwiki-17431 | Determinant of Elementary Column Matrix/Scale Column and Add | Let $e_2$ be the elementary column operation $\text {ECO} 2$:
{{begin-axiom}}
{{axiom | n = \text {ECO} 2
| t = For some $\lambda$, add $\lambda$ times column $j$ to column $i$
| m = \kappa_i \to \kappa_i + \lambda \kappa_j
}}
{{end-axiom}}
which is to operate on some arbitrary matrix space.
Let $\mathb... | By Elementary Matrix corresponding to Elementary Column Operation: Scale Column and Add, $\mathbf E_2$ is of the form:
:$E_{a b} = \delta_{a b} + \lambda \cdot \delta_{b i} \cdot \delta_{j a}$
where:
:$E_{a b}$ denotes the element of $\mathbf E$ whose indices are $\tuple {a, b}$
:$\delta_{a b}$ is the Kronecker delta:
... | Let $e_2$ be the [[Definition:Elementary Column Operation|elementary column operation]] $\text {ECO} 2$:
{{begin-axiom}}
{{axiom | n = \text {ECO} 2
| t = For some $\lambda$, add $\lambda$ [[Definition:Matrix Scalar Product|times]] [[Definition:Column of Matrix|column]] $j$ to [[Definition:Column of Matrix|col... | By [[Elementary Matrix corresponding to Elementary Column Operation/Scale Column and Add|Elementary Matrix corresponding to Elementary Column Operation: Scale Column and Add]], $\mathbf E_2$ is of the form:
:$E_{a b} = \delta_{a b} + \lambda \cdot \delta_{b i} \cdot \delta_{j a}$
where:
:$E_{a b}$ denotes the [[Defini... | Determinant of Elementary Column Matrix/Scale Column and Add | https://proofwiki.org/wiki/Determinant_of_Elementary_Column_Matrix/Scale_Column_and_Add | https://proofwiki.org/wiki/Determinant_of_Elementary_Column_Matrix/Scale_Column_and_Add | [
"Determinant of Elementary Matrix"
] | [
"Definition:Elementary Operation/Column",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix Space",
"Definition:Elementary Matrix/Column Operation",
"Definition:Determinant/Matrix"
] | [
"Elementary Matrix corresponding to Elementary Column Operation/Scale Column and Add",
"Definition:Matrix/Element",
"Definition:Matrix/Indices",
"Definition:Kronecker Delta",
"Definition:Main Diagonal/Diagonal Elements",
"Definition:Matrix/Element",
"Definition:Main Diagonal/Diagonal Elements",
"Defin... |
proofwiki-17432 | Multiple of Column Added to Column of Determinant | Let <nowiki>$\mathbf A = \begin {bmatrix}
a_{1 1} & \cdots & a_{1 r} & \cdots & a_{1 s} & \cdots & a_{1 n} \\
a_{2 1} & \cdots & a_{2 r} & \cdots & a_{2 s} & \cdots & a_{2 n} \\
\vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n 1} & \cdots & a_{n r} & \cdots & a_{n s} & \cdots & a_{n n} \\
\end {... | We have that:
:<nowiki>$\mathbf A^\intercal = \begin {bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{r 1} & a_{r 2} & \cdots & a_{r n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{s 1} & a_{s 2} & \cdots & a_{s n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2}... | Let <nowiki>$\mathbf A = \begin {bmatrix}
a_{1 1} & \cdots & a_{1 r} & \cdots & a_{1 s} & \cdots & a_{1 n} \\
a_{2 1} & \cdots & a_{2 r} & \cdots & a_{2 s} & \cdots & a_{2 n} \\
\vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n 1} & \cdots & a_{n r} & \cdots & a_{n s} & \cdots & a_{n n} \\
\end {... | We have that:
:<nowiki>$\mathbf A^\intercal = \begin {bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{r 1} & a_{r 2} & \cdots & a_{r n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{s 1} & a_{s 2} & \cdots & a_{s n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2... | Multiple of Column Added to Column of Determinant | https://proofwiki.org/wiki/Multiple_of_Column_Added_to_Column_of_Determinant | https://proofwiki.org/wiki/Multiple_of_Column_Added_to_Column_of_Determinant | [
"Determinants"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Determinant/Matrix",
"Definition:Constant",
"Definition:Matrix/Column",
"Definition:Matrix/Column"
] | [
"Definition:Transpose of Matrix",
"Multiple of Row Added to Row of Determinant",
"Determinant of Transpose"
] |
proofwiki-17433 | Determinant with Column Multiplied by Constant | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Let $\mathbf B$ be the matrix resulting from one column of $\mathbf A$ having been multiplied by a constant $c$.
Then:
:$\map \det {\mathbf B} = c \map \det {\mathbf A}$
That is, multiplying one ... | Let:
:<nowiki>$\mathbf A = \begin{bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 r} & \cdots & a_{1 n} \\
a_{2 1} & a_{2 2} & \cdots & a_{2 r} & \cdots & a_{2 n} \\
\vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n r} & \cdots & a_{n n} \\
\end{bmatrix}$</nowiki>
:<nowiki>$\mathbf... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\map \det {\mathbf A}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Let $\mathbf B$ be the [[Definition:Square Matrix|matrix]] resulting from one [[Definiti... | Let:
:<nowiki>$\mathbf A = \begin{bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 r} & \cdots & a_{1 n} \\
a_{2 1} & a_{2 2} & \cdots & a_{2 r} & \cdots & a_{2 n} \\
\vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n r} & \cdots & a_{n n} \\
\end{bmatrix}$</nowiki>
:<nowiki>$\mathb... | Determinant with Column Multiplied by Constant | https://proofwiki.org/wiki/Determinant_with_Column_Multiplied_by_Constant | https://proofwiki.org/wiki/Determinant_with_Column_Multiplied_by_Constant | [
"Determinants"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Column",
"Definition:Constant",
"Definition:Matrix/Column",
"Definition:Matrix/Square Matrix",
"Definition:Constant",
"Definition:Deter... | [
"Definition:Transpose of Matrix",
"Determinant with Row Multiplied by Constant",
"Determinant of Transpose"
] |
proofwiki-17434 | Determinant of Elementary Column Matrix/Exchange Columns | Let $e_3$ be the elementary column operation $\text {ECO} 3$:
{{begin-axiom}}
{{axiom | n = \text {ECO} 3
| t = Exchange columns $i$ and $j$
| m = \kappa_i \leftrightarrow \kappa_j
}}
{{end-axiom}}
which is to operate on some arbitrary matrix space.
Let $\mathbf E_3$ be the elementary column matrix corr... | 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 Column Operation|elementary column operation]] $\text {ECO} 3$:
{{begin-axiom}}
{{axiom | n = \text {ECO} 3
| t = Exchange [[Definition:Column of Matrix|columns]] $i$ and $j$
| m = \kappa_i \leftrightarrow \kappa_j
}}
{{end-axiom}}
which is to operate on some a... | 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 Column Matrix/Exchange Columns | https://proofwiki.org/wiki/Determinant_of_Elementary_Column_Matrix/Exchange_Columns | https://proofwiki.org/wiki/Determinant_of_Elementary_Column_Matrix/Exchange_Columns | [
"Determinant of Elementary Matrix"
] | [
"Definition:Elementary Operation/Column",
"Definition:Matrix/Column",
"Definition:Matrix Space",
"Definition:Elementary Matrix/Column 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/Column",
"Definition:Transposition",
"Definition:Determinant/Matrix",
"Permutation of Determin... |
proofwiki-17435 | Sequence of Row Operations is 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 $\Gamma_1$ be a row operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$.
Let $\Gamma_2$ be a row operation which transforms $\mathbf B$ to another new ... | Let $\sequence {e_i}_{1 \mathop \le i \mathop \le k}$ be the finite sequence of elementary row operations that compose $\Gamma_1$.
Let $\sequence {\mathbf E_i}_{1 \mathop \le i \mathop \le k}$ be the corresponding finite sequence of the elementary row matrices.
Let $\sequence {f_i}_{1 \mathop \le i \mathop \le l}$ be t... | 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 $\Gamma_1$ be a [[Definition:Row Operation|row operation]] whic... | Let $\sequence {e_i}_{1 \mathop \le i \mathop \le k}$ be the [[Definition:Finite Sequence|finite sequence]] of [[Definition:Elementary Row Operation|elementary row operations]] that compose $\Gamma_1$.
Let $\sequence {\mathbf E_i}_{1 \mathop \le i \mathop \le k}$ be the corresponding [[Definition:Finite Sequence|finit... | Sequence of Row Operations is Row Operation | https://proofwiki.org/wiki/Sequence_of_Row_Operations_is_Row_Operation | https://proofwiki.org/wiki/Sequence_of_Row_Operations_is_Row_Operation | [
"Row Operations"
] | [
"Definition:Metric Space",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Matrix",
"Definition:Row Operation",
"Definition:Matrix",
"Definition:Row Operation",
"Definition:Matrix",
"Definition:Row Operation"
] | [
"Definition:Finite Sequence",
"Definition:Elementary Operation/Row",
"Definition:Finite Sequence",
"Definition:Elementary Matrix/Row Operation",
"Definition:Finite Sequence",
"Definition:Elementary Operation/Row",
"Definition:Finite Sequence",
"Definition:Elementary Matrix/Row Operation",
"Row Opera... |
proofwiki-17436 | Sequence of Column Operations is Column 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 $\Gamma_1$ be a column operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$.
Let $\Gamma_2$ be a column operation which transforms $\mathbf B$ to anothe... | Let $\sequence {e_i}_{1 \mathop \le i \mathop \le k}$ be the finite sequence of elementary column operations that compose $\Gamma_1$.
Let $\sequence {\mathbf E_i}_{1 \mathop \le i \mathop \le k}$ be the corresponding finite sequence of the elementary column matrices.
Let $\sequence {f_i}_{1 \mathop \le i \mathop \le l}... | 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 $\Gamma_1$ be a [[Definition:Column Operation|column operation]... | Let $\sequence {e_i}_{1 \mathop \le i \mathop \le k}$ be the [[Definition:Finite Sequence|finite sequence]] of [[Definition:Elementary Column Operation|elementary column operations]] that compose $\Gamma_1$.
Let $\sequence {\mathbf E_i}_{1 \mathop \le i \mathop \le k}$ be the corresponding [[Definition:Finite Sequence... | Sequence of Column Operations is Column Operation | https://proofwiki.org/wiki/Sequence_of_Column_Operations_is_Column_Operation | https://proofwiki.org/wiki/Sequence_of_Column_Operations_is_Column_Operation | [
"Column Operations"
] | [
"Definition:Metric Space",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Matrix",
"Definition:Column Operation",
"Definition:Matrix",
"Definition:Column Operation",
"Definition:Matrix",
"Definition:Column Operation"
] | [
"Definition:Finite Sequence",
"Definition:Elementary Operation/Column",
"Definition:Finite Sequence",
"Definition:Elementary Matrix/Column Operation",
"Definition:Finite Sequence",
"Definition:Elementary Operation/Column",
"Definition:Finite Sequence",
"Definition:Elementary Matrix/Column Operation",
... |
proofwiki-17437 | Equivalence of Definitions of Determinant | Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
{{TFAE|def = Determinant of Matrix|view = the determinant of $\mathbf A$}} | This is proved in Laplace Expansion Theorem for Determinants.
{{qed}}
Category:Determinants
32f91fz73q4e29hefegk5saeroae633 | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
{{TFAE|def = Determinant of Matrix|view = the determinant of $\mathbf A$}} | This is proved in [[Laplace Expansion Theorem for Determinants]].
{{qed}}
[[Category:Determinants]]
32f91fz73q4e29hefegk5saeroae633 | Equivalence of Definitions of Determinant | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Determinant | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Determinant | [
"Determinants"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order"
] | [
"Laplace Expansion Theorem for Determinants",
"Category:Determinants"
] |
proofwiki-17438 | Intersection With Singleton is Disjoint if Not Element | Let $S$ be a set.
Let $\set x$ be the singleton of $x$.
Then:
:$x \notin S$ {{iff}} $\set x \cap S = \O$ | {{begin-eqn}}
{{eqn | o =
| r = \set x \cap S = \O
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \forall y
| o =
| r = \lnot \paren {y \in \set x \cap S}
| c = {{Defof|Empty Set}}
}}
{{eqn | ll= \leadstoandfrom
| q = \forall y
| o =
| r = \lnot \paren {y \in \set x \la... | Let $S$ be a [[Definition:Set|set]].
Let $\set x$ be the [[Definition:Singleton|singleton of $x$]].
Then:
:$x \notin S$ {{iff}} $\set x \cap S = \O$ | {{begin-eqn}}
{{eqn | o =
| r = \set x \cap S = \O
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \forall y
| o =
| r = \lnot \paren {y \in \set x \cap S}
| c = {{Defof|Empty Set}}
}}
{{eqn | ll= \leadstoandfrom
| q = \forall y
| o =
| r = \lnot \paren {y \in \set x \la... | Intersection With Singleton is Disjoint if Not Element | https://proofwiki.org/wiki/Intersection_With_Singleton_is_Disjoint_if_Not_Element | https://proofwiki.org/wiki/Intersection_With_Singleton_is_Disjoint_if_Not_Element | [
"Singletons",
"Disjoint Sets"
] | [
"Definition:Set",
"Definition:Singleton"
] | [
"Double Negation/Double Negation Introduction",
"Conditional is Equivalent to Negation of Conjunction with Negative",
"Category:Singletons",
"Category:Disjoint Sets"
] |
proofwiki-17439 | Determinant of Lower Triangular Matrix | Let $\mathbf T_n$ be a lower 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:
:$\ds \map \det {\mathbf T_n} = \prod_{k \mathop = 1}^n a_{k k}$ | From Transpose of Upper Triangular Matrix is Lower Triangular, the transpose $\mathbf T_n^\intercal$ of $\mathbf T_n$ is an upper triangular matrix.
From Determinant of Upper Triangular Matrix, the determinant of $\mathbf T_n^\intercal$ is equal to the product of all the diagonal elements of $\mathbf T_n^\intercal$.
Fr... | Let $\mathbf T_n$ be a [[Definition:Lower Triangular Matrix|lower 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 [[Definit... | From [[Transpose of Upper Triangular Matrix is Lower Triangular]], the [[Definition:Transpose of Matrix|transpose]] $\mathbf T_n^\intercal$ of $\mathbf T_n$ is an [[Definition:Upper Triangular Matrix|upper triangular matrix]].
From [[Determinant of Upper Triangular Matrix]], the [[Definition:Determinant of Matrix|dete... | Determinant of Lower Triangular Matrix | https://proofwiki.org/wiki/Determinant_of_Lower_Triangular_Matrix | https://proofwiki.org/wiki/Determinant_of_Lower_Triangular_Matrix | [
"Determinants",
"Lower Triangular Matrices"
] | [
"Definition:Triangular Matrix/Lower Triangular Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Main Diagonal/Diagonal Elements"
] | [
"Transpose of Upper Triangular Matrix is Lower Triangular",
"Definition:Transpose of Matrix",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Determinant of Upper Triangular Matrix",
"Definition:Determinant/Matrix",
"Definition:Main Diagonal/Diagonal Elements",
"Determinant of Transpose",
"De... |
proofwiki-17440 | Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant | Let $\mathbf A = \sqbrk a_n$ be a nonsingular square matrix of order $n$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Let $\adj {\mathbf A}$ be the adjugate of $\mathbf A$.
Then:
:$\mathbf A^{-1} = \dfrac 1 {\map \det {\mathbf A} } \cdot \adj {\mathbf A}$
where $\mathbf A^{-1}$ denotes the inverse of... | Let $\mathbf I_n$ denote the unit matrix of order $n$.
{{begin-eqn}}
{{eqn | l = \map \det {\mathbf A} \cdot \mathbf I_n
| r = \mathbf A \cdot \adj {\mathbf A}
| c = Matrix Product with Adjugate Matrix
}}
{{eqn | l = \map \det {\mathbf A} \cdot \mathbf A^{-1} \cdot \mathbf I_n
| r = \mathbf A^{-1} \cd... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Nonsingular Matrix|nonsingular]] [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\map \det {\mathbf A}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Let $\adj {\mathbf A}$ be the [[Definition:... | Let $\mathbf I_n$ denote the [[Definition:Unit Matrix|unit matrix]] of [[Definition:Order of Square Matrix|order $n$]].
{{begin-eqn}}
{{eqn | l = \map \det {\mathbf A} \cdot \mathbf I_n
| r = \mathbf A \cdot \adj {\mathbf A}
| c = [[Matrix Product with Adjugate Matrix]]
}}
{{eqn | l = \map \det {\mathbf A}... | Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant/Proof 1 | https://proofwiki.org/wiki/Inverse_of_Matrix_is_Scalar_Product_of_Adjugate_by_Reciprocal_of_Determinant | https://proofwiki.org/wiki/Inverse_of_Matrix_is_Scalar_Product_of_Adjugate_by_Reciprocal_of_Determinant/Proof_1 | [
"Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant",
"Adjugate Matrices",
"Determinants"
] | [
"Definition:Nonsingular Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Adjugate Matrix",
"Definition:Inverse Matrix"
] | [
"Definition:Unit Matrix",
"Definition:Matrix/Square Matrix/Order",
"Matrix Product with Adjugate Matrix",
"Unit Matrix is Identity for Matrix Multiplication"
] |
proofwiki-17441 | Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant | Let $\mathbf A = \sqbrk a_n$ be a nonsingular square matrix of order $n$.
Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.
Let $\adj {\mathbf A}$ be the adjugate of $\mathbf A$.
Then:
:$\mathbf A^{-1} = \dfrac 1 {\map \det {\mathbf A} } \cdot \adj {\mathbf A}$
where $\mathbf A^{-1}$ denotes the inverse of... | Let:
:$\mathbf A = \begin {bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end {bmatrix}$
:$\mathbf A^{-1} = \begin {bmatrix} b_{11} & \cdots & b_{1n} \\ \vdots & \ddots & \vdots \\ b_{n1} & \cdots & b_{nn} \end {bmatrix}$
Let $\tuple {\mathbf e_1, \mathbf e_2, \cdots, \mathbf... | Let $\mathbf A = \sqbrk a_n$ be a [[Definition:Nonsingular Matrix|nonsingular]] [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\map \det {\mathbf A}$ be the [[Definition:Determinant of Matrix|determinant]] of $\mathbf A$.
Let $\adj {\mathbf A}$ be the [[Definition:... | Let:
:$\mathbf A = \begin {bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \end {bmatrix}$
:$\mathbf A^{-1} = \begin {bmatrix} b_{11} & \cdots & b_{1n} \\ \vdots & \ddots & \vdots \\ b_{n1} & \cdots & b_{nn} \end {bmatrix}$
Let $\tuple {\mathbf e_1, \mathbf e_2, \cdots, \mathb... | Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant/Proof 2 | https://proofwiki.org/wiki/Inverse_of_Matrix_is_Scalar_Product_of_Adjugate_by_Reciprocal_of_Determinant | https://proofwiki.org/wiki/Inverse_of_Matrix_is_Scalar_Product_of_Adjugate_by_Reciprocal_of_Determinant/Proof_2 | [
"Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant",
"Adjugate Matrices",
"Determinants"
] | [
"Definition:Nonsingular Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix",
"Definition:Adjugate Matrix",
"Definition:Inverse Matrix"
] | [
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Linear Transformation/Vector Space",
"Linear Transformation as Matrix Product",
"Definition:Unit Matrix",
"Unit Matrix is Identity for Matrix Multiplication",
"Definition:Matrix",
"Definition:Matrix/Column",
"Definition:Cofactor/Element",
... |
proofwiki-17442 | Greedy Algorithm yields Maximal Set | Let $\struct{S,\mathscr F}$ be an independence system.
Let $w : S \to \R_{\ge 0}$ be a weight function.
Then the Greedy Algorithm selects a maximal set $A_0$ in $\mathscr F$. | {{ProofWanted}}
Category:Maximization Problem for Independence Systems
omhyyhdrmhuqurhcydk3z4mmkhrgbcg | Let $\struct{S,\mathscr F}$ be an [[Definition:Independence System|independence system]].
Let $w : S \to \R_{\ge 0}$ be a [[Definition:Weight Function|weight function]].
Then the [[Maximization Problem (Greedy Algorithm)|Greedy Algorithm]] selects a [[Definition:Maximal Set|maximal set]] $A_0$ in $\mathscr F$. | {{ProofWanted}}
[[Category:Maximization Problem for Independence Systems]]
omhyyhdrmhuqurhcydk3z4mmkhrgbcg | Greedy Algorithm yields Maximal Set | https://proofwiki.org/wiki/Greedy_Algorithm_yields_Maximal_Set | https://proofwiki.org/wiki/Greedy_Algorithm_yields_Maximal_Set | [
"Maximization Problem for Independence Systems"
] | [
"Definition:Independence System",
"Definition:Weight Function",
"Maximization Problem for Independence Systems/Greedy Algorithm",
"Definition:Maximal/Set"
] | [
"Category:Maximization Problem for Independence Systems"
] |
proofwiki-17443 | Greedy Algorithm may not yield Maximum Weight | Let $\struct {S,\mathscr F}$ be an independence system.
Let $w : S \to \R_{\ge 0}$ be a weight function.
Then the maximal set $A_0 \in \mathscr F$ selected by the Greedy Algorithm may not have maximum weight. | {{ProofWanted}}
Category:Maximization Problem for Independence Systems
pvhcoq3lury3y2d5hshriio5ys2rn21 | Let $\struct {S,\mathscr F}$ be an [[Definition:Independence System|independence system]].
Let $w : S \to \R_{\ge 0}$ be a [[Definition:Weight Function|weight function]].
Then the [[Definition:Maximal Set|maximal set]] $A_0 \in \mathscr F$ selected by the [[Maximization Problem (Greedy Algorithm)|Greedy Algorithm]] ... | {{ProofWanted}}
[[Category:Maximization Problem for Independence Systems]]
pvhcoq3lury3y2d5hshriio5ys2rn21 | Greedy Algorithm may not yield Maximum Weight | https://proofwiki.org/wiki/Greedy_Algorithm_may_not_yield_Maximum_Weight | https://proofwiki.org/wiki/Greedy_Algorithm_may_not_yield_Maximum_Weight | [
"Maximization Problem for Independence Systems"
] | [
"Definition:Independence System",
"Definition:Weight Function",
"Definition:Maximal/Set",
"Maximization Problem for Independence Systems/Greedy Algorithm",
"Definition:Maximum Value of Real Function",
"Definition:Extended Weight Function"
] | [
"Category:Maximization Problem for Independence Systems"
] |
proofwiki-17444 | Independent Subset is Contained in Base | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\mathscr B$ denote the set of all bases of $M$.
Let $A \in \mathscr I$.
Then:
:$\exists B \in \mathscr B : A \subseteq B$ | Consider the ordered set $\struct {\mathscr I, \subseteq}$.
From Element of Finite Ordered Set is Between Maximal and Minimal Elements:
:$\exists B \in \mathscr I : A \subseteq B$ and $B$ is maximal in $\struct {\mathscr I, \subseteq}$.
By definition of a base:
:$B \in \mathscr B$
{{qed}}
Category:Matroid Independent S... | 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 $A \in \mathscr I$.
Then:
:$\exists B \in \mathscr B : A \subseteq B$ | Consider the [[Definition:Ordered Set|ordered set]] $\struct {\mathscr I, \subseteq}$.
From [[Element of Finite Ordered Set is Between Maximal and Minimal Elements]]:
:$\exists B \in \mathscr I : A \subseteq B$ and $B$ is [[Definition:Maximal Element|maximal]] in $\struct {\mathscr I, \subseteq}$.
By definition of a ... | Independent Subset is Contained in Base | https://proofwiki.org/wiki/Independent_Subset_is_Contained_in_Base | https://proofwiki.org/wiki/Independent_Subset_is_Contained_in_Base | [
"Matroid Independent Subsets",
"Matroid Bases"
] | [
"Definition:Matroid",
"Definition:Base of Matroid"
] | [
"Definition:Ordered Set",
"Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements/Corollary",
"Definition:Maximal/Element",
"Definition:Base of Matroid",
"Category:Matroid Independent Subsets",
"Category:Matroid Bases"
] |
proofwiki-17445 | Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements/Corollary | Let $\struct {S, \preceq}$ be a finite ordered set.
Let $x \in S$.
Then there exists a maximal element $M \in S$ and a minimal element $m \in S$ such that:
:$m \preceq x \preceq M$ | Let $T = \set{y : x \preceq y}$.
By the reflexivity of the ordering $\preceq$:
:$x \preceq x$
So $x \in T$ and $T$ is non-empty.
From Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements:
:$\struct {T, \preceq}$ has a maximal element $M \in T$
We now show that $M$ is a maximal element in $\struct{S, ... | Let $\struct {S, \preceq}$ be a [[Definition:Finite Set|finite]] [[Definition:Ordered Set|ordered set]].
Let $x \in S$.
Then there exists a [[Definition:Maximal Element|maximal element]] $M \in S$ and a [[Definition:Minimal Element|minimal element]] $m \in S$ such that:
:$m \preceq x \preceq M$ | Let $T = \set{y : x \preceq y}$.
By the [[Definition:Reflexive Relation|reflexivity]] of the [[Definition:Ordering|ordering]] $\preceq$:
:$x \preceq x$
So $x \in T$ and $T$ is [[Definition:Non-Empty Set|non-empty]].
From [[Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements]]:
:$\struct {T, \prec... | Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements/Corollary | https://proofwiki.org/wiki/Finite_Non-Empty_Subset_of_Ordered_Set_has_Maximal_and_Minimal_Elements/Corollary | https://proofwiki.org/wiki/Finite_Non-Empty_Subset_of_Ordered_Set_has_Maximal_and_Minimal_Elements/Corollary | [
"Minimal Elements",
"Maximal Elements"
] | [
"Definition:Finite Set",
"Definition:Ordered Set",
"Definition:Maximal/Element",
"Definition:Minimal/Element"
] | [
"Definition:Reflexive Relation",
"Definition:Ordering",
"Definition:Non-Empty Set",
"Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements",
"Definition:Maximal/Element",
"Definition:Maximal/Element",
"Definition:Transitive Relation",
"Definition:Ordering",
"Definition:Maximal/Elem... |
proofwiki-17446 | Equivalent Conditions for Element is Loop | Let $M = \struct{S, \mathscr I}$ be a matroid.
Let $\sigma$ denote the closure operator on $M$.
Let $\rho$ denote the rank function of $M$.
Let $\mathscr B$ denote the set of all bases of $M$.
Let $x \in S$.
{{TFAE}}
:$(1)\quad x$ is a loop
:$(2)\quad x \in \map \sigma \O$
:$(3)\quad \map \rho {\set x} = 0$
:$(4)\quad... | === Condition $(1)$ iff Condition $(2)$ ===
Follows immediately from Element is Loop iff Member of Closure of Empty Set.
{{qed|lemma}} | Let $M = \struct{S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\sigma$ denote the [[Definition:Closure Operator (Matroid)|closure operator]] on $M$.
Let $\rho$ denote the [[Definition:Rank Function (Matroid)|rank function]] of $M$.
Let $\mathscr B$ denote the set of all [[Definition:Base of Matroid|bases... | === Condition $(1)$ iff Condition $(2)$ ===
Follows immediately from [[Element is Loop iff Member of Closure of Empty Set]].
{{qed|lemma}} | Equivalent Conditions for Element is Loop | https://proofwiki.org/wiki/Equivalent_Conditions_for_Element_is_Loop | https://proofwiki.org/wiki/Equivalent_Conditions_for_Element_is_Loop | [
"Matroid Loops"
] | [
"Definition:Matroid",
"Definition:Closure Operator (Matroid)",
"Definition:Rank Function (Matroid)",
"Definition:Base of Matroid",
"Definition:Loop (Matroid)",
"Definition:Circuit (Matroid)",
"Definition:Element"
] | [
"Element is Loop iff Member of Closure of Empty Set"
] |
proofwiki-17447 | Power Set of Doubleton | Let $x, y$ be distinct objects.
Then the power set of the doubleton $\set {x, y}$ is:
:$\powerset {\set {x, y} } = \set {\O, \set x, \set y, \set {x, y} }$ | By definition of a subset:
:$\set x , \set y, \set {x, y} \subseteq \set {x, y}$
Let $A \subseteq \set {x, y}$:
:$A \ne \set x, \set y, \set {x, y}$
From set equality:
:$\set {x, y} \nsubseteq A$
From Doubleton of Elements is Subset:
:either $x \notin A$ or $y \notin A$.
{{WLOG}} assume that $x \notin A$.
From Intersec... | Let $x, y$ be [[Definition:Distinct|distinct]] [[Definition:Object|objects]].
Then the [[Definition:Power Set|power set]] of the [[Definition:Doubleton|doubleton]] $\set {x, y}$ is:
:$\powerset {\set {x, y} } = \set {\O, \set x, \set y, \set {x, y} }$ | By definition of a [[Definition:Subset|subset]]:
:$\set x , \set y, \set {x, y} \subseteq \set {x, y}$
Let $A \subseteq \set {x, y}$:
:$A \ne \set x, \set y, \set {x, y}$
From [[Definition:Set Equality|set equality]]:
:$\set {x, y} \nsubseteq A$
From [[Doubleton of Elements is Subset]]:
:either $x \notin A$ or $y ... | Power Set of Doubleton | https://proofwiki.org/wiki/Power_Set_of_Doubleton | https://proofwiki.org/wiki/Power_Set_of_Doubleton | [
"Power Set",
"Doubletons"
] | [
"Definition:Distinct",
"Definition:Object",
"Definition:Power Set",
"Definition:Doubleton"
] | [
"Definition:Subset",
"Definition:Set Equality",
"Doubleton of Elements is Subset",
"Intersection With Singleton is Disjoint if Not Element",
"Subset of Set Difference iff Disjoint Set",
"Set Difference of Doubleton and Singleton is Singleton",
"Definition:Set Equality",
"Singleton of Element is Subset... |
proofwiki-17448 | Doubleton of Elements is Subset | Let $S$ be a set.
Let $\set {x,y}$ be the doubleton of distinct $x$ and $y$.
Then:
:$x, y \in S \iff \set {x,y} \subseteq S$ | === Necessary Condition ===
Let $x, y \in S$.
From Singleton of Element is Subset:
:$\set x \subseteq S$
:$\set y \subseteq S$
From Union of Subsets is Subset:
:$\set x \cup \set y \subseteq S$
From Union of Disjoint Singletons is Doubleton:
: $\set x \cup \set y = \set {x, y}$
Hence:
:$\set {x,y} \subseteq S$
{{qed|le... | Let $S$ be a [[Definition:Set|set]].
Let $\set {x,y}$ be the [[Definition:Doubleton|doubleton]] of distinct $x$ and $y$.
Then:
:$x, y \in S \iff \set {x,y} \subseteq S$ | === Necessary Condition ===
Let $x, y \in S$.
From [[Singleton of Element is Subset]]:
:$\set x \subseteq S$
:$\set y \subseteq S$
From [[Union of Subsets is Subset]]:
:$\set x \cup \set y \subseteq S$
From [[Union of Disjoint Singletons is Doubleton]]:
: $\set x \cup \set y = \set {x, y}$
Hence:
:$\set {x,y} \sub... | Doubleton of Elements is Subset | https://proofwiki.org/wiki/Doubleton_of_Elements_is_Subset | https://proofwiki.org/wiki/Doubleton_of_Elements_is_Subset | [
"Subsets",
"Doubletons"
] | [
"Definition:Set",
"Definition:Doubleton"
] | [
"Singleton of Element is Subset",
"Union of Subsets is Subset",
"Union of Disjoint Singletons is Doubleton"
] |
proofwiki-17449 | Sum of Unitary Divisors of Power of Prime | Let $n = p^k$ be the power of a prime number $p$.
Then the sum of all positive unitary divisors of $n$ is $1 + n$. | Let $d \divides n$.
By Divisors of Power of Prime, $d = p^a$ for some positive integer $a \le k$.
We have $\dfrac n d = p^{k - a}$.
Suppose $d$ is a unitary divisor of $n$.
Then $d$ and $\dfrac n d$ are coprime.
If both $a, k - a \ne 0$, $p^a$ and $p^{k - a}$ have a common divisor: $p$.
Hence either $a = 0$ or $k - a =... | Let $n = p^k$ be the [[Definition:Power (Algebra)|power]] of a [[Definition:Prime Number|prime number]] $p$.
Then the sum of all [[Definition:Positive Integer|positive]] [[Definition:Unitary Divisor|unitary divisors]] of $n$ is $1 + n$. | Let $d \divides n$.
By [[Divisors of Power of Prime]], $d = p^a$ for some [[Definition:Positive Integer|positive integer]] $a \le k$.
We have $\dfrac n d = p^{k - a}$.
Suppose $d$ is a [[Definition:Unitary Divisor|unitary divisor]] of $n$.
Then $d$ and $\dfrac n d$ are [[Definition:Coprime Integers|coprime]].
If ... | Sum of Unitary Divisors of Power of Prime | https://proofwiki.org/wiki/Sum_of_Unitary_Divisors_of_Power_of_Prime | https://proofwiki.org/wiki/Sum_of_Unitary_Divisors_of_Power_of_Prime | [
"Prime Numbers",
"Sum of Unitary Divisors"
] | [
"Definition:Power (Algebra)",
"Definition:Prime Number",
"Definition:Positive/Integer",
"Definition:Unitary Divisor"
] | [
"Divisors of Power of Prime",
"Definition:Positive/Integer",
"Definition:Unitary Divisor",
"Definition:Coprime/Integers",
"Definition:Common Divisor",
"Definition:Positive/Integer",
"Definition:Unitary Divisor",
"Category:Prime Numbers",
"Category:Sum of Unitary Divisors"
] |
proofwiki-17450 | Sum of Unitary Divisors is Multiplicative | Let $\map {\sigma^*} n$ denote the sum of unitary divisors of $n$.
Then the function:
:$\ds \sigma^*: \Z_{>0} \to \Z_{>0}: \map {\sigma^*} n = \sum_{\substack d \mathop \divides n \\ d \mathop \perp \frac n d} d$
is multiplicative. | Let $a, b$ be coprime integers.
Because $a$ and $b$ have no common divisor, the divisors of $a b$ are integers of the form $a_i b_j$, where $a_i$ is a divisor of $a$ and $b_j$ is a divisor of $b$.
That is, any divisor $d$ of $a b$ is in the form:
:$d = a_i b_j$
in a unique way, where $a_i \divides a$ and $b_j \divides ... | Let $\map {\sigma^*} n$ denote the sum of [[Definition:Unitary Divisor|unitary divisors]] of $n$.
Then the function:
:$\ds \sigma^*: \Z_{>0} \to \Z_{>0}: \map {\sigma^*} n = \sum_{\substack d \mathop \divides n \\ d \mathop \perp \frac n d} d$
is [[Definition:Multiplicative Arithmetic Function|multiplicative]]. | Let $a, b$ be [[Definition:Coprime Integers|coprime integers]].
Because $a$ and $b$ have no [[Definition:Common Divisor of Integers|common divisor]], the [[Definition:Divisor of Integer|divisors]] of $a b$ are [[Definition:Integer|integers]] of the form $a_i b_j$, where $a_i$ is a [[Definition:Divisor of Integer|divis... | Sum of Unitary Divisors is Multiplicative | https://proofwiki.org/wiki/Sum_of_Unitary_Divisors_is_Multiplicative | https://proofwiki.org/wiki/Sum_of_Unitary_Divisors_is_Multiplicative | [
"Sum of Unitary Divisors",
"Multiplicative Functions"
] | [
"Definition:Unitary Divisor",
"Definition:Multiplicative Arithmetic Function"
] | [
"Definition:Coprime/Integers",
"Definition:Common Divisor/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Unique",
"Definition:Unitary Divisor"... |
proofwiki-17451 | Sum of Unitary Divisors of Integer | Let $n$ be an integer such that $n \ge 2$.
Let $\map {\sigma^*} n$ be the sum of all positive unitary divisors of $n$.
Let the prime decomposition of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$
Then:
:$\ds \map {\sigma^*} n = \prod_{1 \mathop \le i \mathop ... | We have that the Sum of Unitary Divisors is Multiplicative.
From Value of Multiplicative Function is Product of Values of Prime Power Factors, we have:
:$\map {\sigma^*} n = \map {\sigma^*} {p_1^{k_1} } \map {\sigma^*} {p_2^{k_2} } \ldots \map {\sigma^*} {p_r^{k_r} }$
From Sum of Unitary Divisors of Power of Prime, we ... | Let $n$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
Let $\map {\sigma^*} n$ be the sum of all positive [[Definition:Unitary Divisor|unitary divisors]] of $n$.
Let the [[Definition:Prime Decomposition|prime decomposition]] of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} = p_1^{k_1} ... | We have that the [[Sum of Unitary Divisors is Multiplicative]].
From [[Value of Multiplicative Function is Product of Values of Prime Power Factors]], we have:
:$\map {\sigma^*} n = \map {\sigma^*} {p_1^{k_1} } \map {\sigma^*} {p_2^{k_2} } \ldots \map {\sigma^*} {p_r^{k_r} }$
From [[Sum of Unitary Divisors of Power o... | Sum of Unitary Divisors of Integer | https://proofwiki.org/wiki/Sum_of_Unitary_Divisors_of_Integer | https://proofwiki.org/wiki/Sum_of_Unitary_Divisors_of_Integer | [
"Sum of Unitary Divisors"
] | [
"Definition:Integer",
"Definition:Unitary Divisor",
"Definition:Prime Decomposition"
] | [
"Sum of Unitary Divisors is Multiplicative",
"Value of Multiplicative Function is Product of Values of Prime Power Factors",
"Sum of Unitary Divisors of Power of Prime",
"Category:Sum of Unitary Divisors"
] |
proofwiki-17452 | Range of Infinite Sequence may be Finite | Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence.
Then it is possible for the range of $\sequence {x_n}$ to be finite. | Consider the infinite sequence $\sequence {x_n}_{n \mathop \in \N}$ defined as:
:$\forall n \in \N: x_n = \dfrac {1 + \paren {-1}^n} 2$
Thus:
:$\sequence {x_n}_{n \mathop \in \N} = 1, 0, 1, 0, \dotsc$
Hence the range of $\sequence {x_n}$ is $\set {0, 1}$, which is finite.
{{qed}} | Let $\sequence {x_n}_{n \mathop \in \N}$ be an [[Definition:Infinite Sequence|infinite sequence]].
Then it is possible for the [[Definition:Range of Sequence|range]] of $\sequence {x_n}$ to be [[Definition:Finite Set|finite]]. | Consider the [[Definition:Infinite Sequence|infinite sequence]] $\sequence {x_n}_{n \mathop \in \N}$ defined as:
:$\forall n \in \N: x_n = \dfrac {1 + \paren {-1}^n} 2$
Thus:
:$\sequence {x_n}_{n \mathop \in \N} = 1, 0, 1, 0, \dotsc$
Hence the [[Definition:Range of Sequence|range]] of $\sequence {x_n}$ is $\set {0, ... | Range of Infinite Sequence may be Finite | https://proofwiki.org/wiki/Range_of_Infinite_Sequence_may_be_Finite | https://proofwiki.org/wiki/Range_of_Infinite_Sequence_may_be_Finite | [
"Sequences"
] | [
"Definition:Sequence/Infinite Sequence",
"Definition:Range of Sequence",
"Definition:Finite Set"
] | [
"Definition:Sequence/Infinite Sequence",
"Definition:Range of Sequence",
"Definition:Finite Set"
] |
proofwiki-17453 | Squares Ending in n Occurrences of m-Digit Pattern | Suppose there exists some integer $x$ such that $x^2$ ends in some $m$-digit pattern ending in an odd number not equal to $5$ and is preceded by another odd number, i.e.:
:$\exists x \in \Z: x^2 \equiv \sqbrk {1 a_1 a_2 \cdots a_m} \pmod {2 \times 10^m}$
where $a_m$ is odd, $a_m \ne 5$ and $m \ge 1$.
Then for any $n \g... | We prove that there exists a sequence $\sequence {b_n}$ with the properties:
:$b_n < 10^{m n}$
:$b_n^2 \equiv \underbrace {\sqbrk {1 \paren {a_1 \cdots a_m} \cdots \paren {a_1 \cdots a_m}}}_{n \text { occurrences}} \pmod {2 \times 10^{m n}}$
by induction: | Suppose there exists some [[Definition:Integer|integer]] $x$ such that $x^2$ ends in some $m$-[[Definition:Digit|digit]] pattern ending in an [[Definition:Odd Integer|odd number]] not equal to $5$ and is preceded by another [[Definition:Odd Integer|odd number]], i.e.:
:$\exists x \in \Z: x^2 \equiv \sqbrk {1 a_1 a_2 \c... | We prove that there exists a [[Definition:Integer Sequence|sequence]] $\sequence {b_n}$ with the properties:
:$b_n < 10^{m n}$
:$b_n^2 \equiv \underbrace {\sqbrk {1 \paren {a_1 \cdots a_m} \cdots \paren {a_1 \cdots a_m}}}_{n \text { occurrences}} \pmod {2 \times 10^{m n}}$
by [[Principle of Mathematical Induction|indu... | Squares Ending in n Occurrences of m-Digit Pattern | https://proofwiki.org/wiki/Squares_Ending_in_n_Occurrences_of_m-Digit_Pattern | https://proofwiki.org/wiki/Squares_Ending_in_n_Occurrences_of_m-Digit_Pattern | [
"Number Theory",
"Recreational Mathematics",
"Squares Ending in n Occurrences of m-Digit Pattern"
] | [
"Definition:Integer",
"Definition:Digit",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Integer",
"Definition:Digit",
"Definition:Square Number",
"Definition:Digit",
"Definition:Odd Integer",
"Definition:Square Number",
"Definition:Digit"
] | [
"Definition:Integer Sequence",
"Principle of Mathematical Induction",
"Definition:Integer",
"Principle of Mathematical Induction"
] |
proofwiki-17454 | Largest Number not Expressible as Sum of Multiples of Coprime Integers | Let $a, b$ be coprime integers, each greater than $1$.
Then the largest number not expressible as a sum of multiples of $a$ and $b$ (possibly zero) is the number:
:$a b - a - b = \paren {a - 1} \paren {b - 1} - 1$ | First we show that $a b - a - b$ is not expressible as a sum of multiples of $a$ and $b$.
{{AimForCont}} $a b - a - b = s a + t b$ for some $s, t \in \N$.
Note that $t b \le s a + t b < a b - b = \paren {a - 1} b$.
This gives $t < a - 1$.
We also have $\paren {a - t - 1} b = \paren {s + 1} a$.
Hence $a \divides \paren ... | Let $a, b$ be [[Definition:Coprime Integers|coprime integers]], each greater than $1$.
Then the largest number not expressible as a sum of multiples of $a$ and $b$ (possibly zero) is the number:
:$a b - a - b = \paren {a - 1} \paren {b - 1} - 1$ | First we show that $a b - a - b$ is not expressible as a sum of multiples of $a$ and $b$.
{{AimForCont}} $a b - a - b = s a + t b$ for some $s, t \in \N$.
Note that $t b \le s a + t b < a b - b = \paren {a - 1} b$.
This gives $t < a - 1$.
We also have $\paren {a - t - 1} b = \paren {s + 1} a$.
Hence $a \divides \... | Largest Number not Expressible as Sum of Multiples of Coprime Integers | https://proofwiki.org/wiki/Largest_Number_not_Expressible_as_Sum_of_Multiples_of_Coprime_Integers | https://proofwiki.org/wiki/Largest_Number_not_Expressible_as_Sum_of_Multiples_of_Coprime_Integers | [
"Largest Number not Expressible as Sum of Multiples of Coprime Integers",
"Frobenius Numbers",
"Coin Problem",
"Integer Combinations"
] | [
"Definition:Coprime/Integers"
] | [
"Definition:Coprime/Integers",
"Euclid's Lemma",
"Absolute Value of Integer is not less than Divisors",
"Definition:Contradiction",
"Absolute Value of Integer is not less than Divisors",
"Euclid's Lemma",
"Absolute Value of Integer is not less than Divisors"
] |
proofwiki-17455 | Fermat Quotient of 2 wrt p is Square iff p is 3 or 7/Generalization | The Fermat quotient of $2$ with respect to $p$:
:$\map {q_p} 2 = \dfrac {2^{p - 1} - 1} p$
is a perfect power {{iff}} $p = 3$ or $p = 7$. | To show that these are the only ones, we observe that since $p$ is an odd prime, write:
:$p = 2 n + 1$ for $n \ge 1$.
Let $\map {q_p} 2$ be a perfect power.
Then $2^{p - 1} - 1 = p x^y$ for some integers $x, y$.
Note that:
:$2^{p - 1} - 1 = 2^{2 n} - 1 = \paren {2^n - 1} \paren {2^n + 1}$
and we have:
:$\gcd \set {2^n ... | The [[Definition:Fermat Quotient|Fermat quotient]] of $2$ with respect to $p$:
:$\map {q_p} 2 = \dfrac {2^{p - 1} - 1} p$
is a [[Definition:Perfect Power|perfect power]] {{iff}} $p = 3$ or $p = 7$. | To show that these are the only ones, we observe that since $p$ is an [[Definition:Odd Prime|odd prime]], write:
:$p = 2 n + 1$ for $n \ge 1$.
Let $\map {q_p} 2$ be a [[Definition:Perfect Power|perfect power]].
Then $2^{p - 1} - 1 = p x^y$ for some [[Definition:Integer|integers]] $x, y$.
Note that:
:$2^{p - 1} - 1... | Fermat Quotient of 2 wrt p is Square iff p is 3 or 7/Generalization | https://proofwiki.org/wiki/Fermat_Quotient_of_2_wrt_p_is_Square_iff_p_is_3_or_7/Generalization | https://proofwiki.org/wiki/Fermat_Quotient_of_2_wrt_p_is_Square_iff_p_is_3_or_7/Generalization | [
"Fermat Quotients"
] | [
"Definition:Fermat Quotient",
"Definition:Perfect Power"
] | [
"Definition:Odd Prime",
"Definition:Perfect Power",
"Definition:Integer",
"Definition:Coprime/Integers",
"Definition:Perfect Power",
"Definition:Perfect Power",
"Definition:Perfect Power"
] |
proofwiki-17456 | Taxicab Norm is Norm | The taxicab norm is a norm on the real and complex numbers. | By P-Norm is Norm, $\norm {\, \cdot \,}_p$ is a norm.
By definition, the taxicab norm is $\norm {\, \cdot \,}_1$.
Therefore, the taxicab norm is a norm.
{{qed}} | The [[Definition:Taxicab Norm|taxicab norm]] is a [[Definition:Norm on Vector Space|norm]] on the [[Definition:Real Number|real]] and [[Definition:Complex Number|complex numbers]]. | By [[P-Norm is Norm]], $\norm {\, \cdot \,}_p$ is a [[Definition:Norm on Vector Space|norm]].
By definition, the [[Definition:Taxicab Norm|taxicab norm]] is $\norm {\, \cdot \,}_1$.
Therefore, the [[Definition:Taxicab Norm|taxicab norm]] is a [[Definition:Norm on Vector Space|norm]].
{{qed}} | Taxicab Norm is Norm | https://proofwiki.org/wiki/Taxicab_Norm_is_Norm | https://proofwiki.org/wiki/Taxicab_Norm_is_Norm | [
"Taxicab Norm"
] | [
"Definition:Taxicab Norm",
"Definition:Norm/Vector Space",
"Definition:Real Number",
"Definition:Complex Number"
] | [
"P-Norm is Norm",
"Definition:Norm/Vector Space",
"Definition:Taxicab Norm",
"Definition:Taxicab Norm",
"Definition:Norm/Vector Space"
] |
proofwiki-17457 | Multiplication by 2 over 3 in Egyptian Fractions | Let $\dfrac 1 n$ be an Egyptian fraction not equal to $\dfrac 2 3$.
In order to multiply $\dfrac 1 n$ by $\dfrac 2 3$ and have it that $\dfrac 1 n \times \dfrac 2 3$ is also expressed in Egyptian form, we have:
:$\dfrac 1 n \times \dfrac 2 3 = \dfrac 1 {2 n} + \dfrac 1 {6 n}$ | {{begin-eqn}}
{{eqn | l = \dfrac 1 {2 n} + \dfrac 1 {6 n}
| r = \dfrac 3 {6 n} + \dfrac 1 {6 n}
| c =
}}
{{eqn | r = \dfrac {3 + 1} {6 n}
| c =
}}
{{eqn | r = \dfrac 2 {3 n}
| c =
}}
{{eqn | r = \dfrac 1 n \times \dfrac 2 3
| c =
}}
{{end-eqn}}
Note the case where we multiply $\dfrac 2... | Let $\dfrac 1 n$ be an [[Definition:Egyptian Fraction|Egyptian fraction]] not equal to $\dfrac 2 3$.
In order to [[Definition:Rational Multiplication|multiply]] $\dfrac 1 n$ by $\dfrac 2 3$ and have it that $\dfrac 1 n \times \dfrac 2 3$ is also expressed in [[Definition:Egyptian Fraction|Egyptian form]], we have:
:... | {{begin-eqn}}
{{eqn | l = \dfrac 1 {2 n} + \dfrac 1 {6 n}
| r = \dfrac 3 {6 n} + \dfrac 1 {6 n}
| c =
}}
{{eqn | r = \dfrac {3 + 1} {6 n}
| c =
}}
{{eqn | r = \dfrac 2 {3 n}
| c =
}}
{{eqn | r = \dfrac 1 n \times \dfrac 2 3
| c =
}}
{{end-eqn}}
Note the case where we multiply $\dfrac... | Multiplication by 2 over 3 in Egyptian Fractions | https://proofwiki.org/wiki/Multiplication_by_2_over_3_in_Egyptian_Fractions | https://proofwiki.org/wiki/Multiplication_by_2_over_3_in_Egyptian_Fractions | [
"Egyptian Fractions",
"Multiplication by 2 over 3 in Egyptian Fractions"
] | [
"Definition:Egyptian Fraction",
"Definition:Multiplication/Rational Numbers",
"Definition:Egyptian Fraction"
] | [
"Definition:Egyptian Fraction"
] |
proofwiki-17458 | Upper Limit of Number of Unit Fractions to express Proper Fraction from Greedy Algorithm | Let $\dfrac p q$ denote a proper fraction expressed in canonical form.
Let $\dfrac p q$ be expressed as the sum of a finite number of distinct unit fractions using Fibonacci's Greedy Algorithm.
Then $\dfrac p q$ is expressed using no more than $p$ unit fractions. | Let $\dfrac {x_k} {y_k}$ and $\dfrac {x_{k + 1} } {y_{k + 1} }$ be consecutive stages of the calculation of the unit fractions accordingly:
:$\dfrac {x_k} {y_k} - \dfrac 1 {\ceiling {y_n / x_n} } = \dfrac {x_{k + 1} } {y_{k + 1} }$
By definition of Fibonacci's Greedy Algorithm:
:$\dfrac {x_{k + 1} } {y_{k + 1} } = \dfr... | Let $\dfrac p q$ denote a [[Definition:Proper Fraction|proper fraction]] expressed in [[Definition:Canonical Form of Rational Number|canonical form]].
Let $\dfrac p q$ be expressed as the [[Definition:Integer Addition|sum]] of a [[Definition:Finite Set|finite number]] of [[Definition:Distinct Elements|distinct]] [[Def... | Let $\dfrac {x_k} {y_k}$ and $\dfrac {x_{k + 1} } {y_{k + 1} }$ be consecutive stages of the calculation of the [[Definition:Unit Fraction|unit fractions]] accordingly:
:$\dfrac {x_k} {y_k} - \dfrac 1 {\ceiling {y_n / x_n} } = \dfrac {x_{k + 1} } {y_{k + 1} }$
By definition of [[Fibonacci's Greedy Algorithm]]:
:$\df... | Upper Limit of Number of Unit Fractions to express Proper Fraction from Greedy Algorithm | https://proofwiki.org/wiki/Upper_Limit_of_Number_of_Unit_Fractions_to_express_Proper_Fraction_from_Greedy_Algorithm | https://proofwiki.org/wiki/Upper_Limit_of_Number_of_Unit_Fractions_to_express_Proper_Fraction_from_Greedy_Algorithm | [
"Fibonacci's Greedy Algorithm"
] | [
"Definition:Fraction/Proper",
"Definition:Rational Number/Canonical Form",
"Definition:Addition/Integers",
"Definition:Finite Set",
"Definition:Distinct/Plural",
"Definition:Unit Fraction",
"Fibonacci's Greedy Algorithm",
"Definition:Unit Fraction"
] | [
"Definition:Unit Fraction",
"Fibonacci's Greedy Algorithm",
"Fibonacci's Greedy Algorithm",
"Definition:Fraction/Numerator",
"Definition:Unit Fraction",
"Definition:Natural Numbers"
] |
proofwiki-17459 | Smallest n for which 3 over n produces 3 Egyptian Fractions using Greedy Algorithm when 2 Sufficient | Consider proper fractions of the form $\dfrac 3 n$ expressed in canonical form.
Let Fibonacci's Greedy Algorithm be used to generate a sequence $S$ of Egyptian fractions for $\dfrac 3 n$.
The smallest $n$ for which $S$ consists of $3$ terms, where $2$ would be sufficient, is $25$. | We have that:
{{begin-eqn}}
{{eqn | l = \frac 3 {25}
| r = \frac 1 9 + \frac 2 {225}
| c = as $\ceiling {25 / 3} = \ceiling {8.333\ldots} = 9$
}}
{{eqn | r = \frac 1 9 + \frac 1 {113} + \frac 1 {25 \, 425}
| c = as $\ceiling {225 / 2} = \ceiling {112.5} = 113$
}}
{{end-eqn}}
But then we have:
{{begin-... | Consider [[Definition:Proper Fraction|proper fractions]] of the form $\dfrac 3 n$ expressed in [[Definition:Canonical Form of Rational Number|canonical form]].
Let [[Fibonacci's Greedy Algorithm]] be used to generate a [[Definition:Sequence|sequence]] $S$ of [[Definition:Egyptian Fraction|Egyptian fractions]] for $\df... | We have that:
{{begin-eqn}}
{{eqn | l = \frac 3 {25}
| r = \frac 1 9 + \frac 2 {225}
| c = as $\ceiling {25 / 3} = \ceiling {8.333\ldots} = 9$
}}
{{eqn | r = \frac 1 9 + \frac 1 {113} + \frac 1 {25 \, 425}
| c = as $\ceiling {225 / 2} = \ceiling {112.5} = 113$
}}
{{end-eqn}}
But then we have:
{{be... | Smallest n for which 3 over n produces 3 Egyptian Fractions using Greedy Algorithm when 2 Sufficient | https://proofwiki.org/wiki/Smallest_n_for_which_3_over_n_produces_3_Egyptian_Fractions_using_Greedy_Algorithm_when_2_Sufficient | https://proofwiki.org/wiki/Smallest_n_for_which_3_over_n_produces_3_Egyptian_Fractions_using_Greedy_Algorithm_when_2_Sufficient | [
"Fibonacci's Greedy Algorithm",
"25"
] | [
"Definition:Fraction/Proper",
"Definition:Rational Number/Canonical Form",
"Fibonacci's Greedy Algorithm",
"Definition:Sequence",
"Definition:Egyptian Fraction",
"Definition:Term of Sequence"
] | [
"Condition for 3 over n producing 3 Egyptian Fractions using Greedy Algorithm when 2 Sufficient",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-17460 | Supremum Norm is Norm/Continuous on Closed Interval Real-Valued Function | Let $I = \closedint a b$ be a closed interval.
Let $\struct {\map C I, +, \, \cdot \,}_\R$ be the vector space of real-valued functions, continuous on $I$.
Let $\map x t \in \map C I$ be a continuous real function.
Let $\size {\, \cdot \,}$ be the absolute value.
Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on... | === Positive definiteness ===
{{begin-eqn}}
{{eqn | l = \norm x_\infty
| r = \sup_{t \mathop \in I} \size {\map x t}
| c = {{Defof|Supremum Norm on Space of Continuous on Closed Interval Real-Valued Functions}}
}}
{{eqn | r = \max_{t \mathop \in I} \size {\map x t}
| c = Weierstrass Extreme Value Theo... | Let $I = \closedint a b$ be a [[Definition:Closed Interval|closed interval]].
Let $\struct {\map C I, +, \, \cdot \,}_\R$ be the [[Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space|vector space of real-valued functions, continuou... | === Positive definiteness ===
{{begin-eqn}}
{{eqn | l = \norm x_\infty
| r = \sup_{t \mathop \in I} \size {\map x t}
| c = {{Defof|Supremum Norm on Space of Continuous on Closed Interval Real-Valued Functions}}
}}
{{eqn | r = \max_{t \mathop \in I} \size {\map x t}
| c = [[Weierstrass Extreme Value T... | Supremum Norm is Norm/Continuous on Closed Interval Real-Valued Function | https://proofwiki.org/wiki/Supremum_Norm_is_Norm/Continuous_on_Closed_Interval_Real-Valued_Function | https://proofwiki.org/wiki/Supremum_Norm_is_Norm/Continuous_on_Closed_Interval_Real-Valued_Function | [
"Examples of Norms",
"Supremum Norm is Norm"
] | [
"Definition:Interval/Ordered Set/Closed",
"Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Definition:Continuous Real Function/Subset",
"Definition:Absolute Value",
"Definition:Supremum Norm/Continuous on Closed I... | [
"Weierstrass Extreme Value Theorem",
"Complex Modulus is Non-Negative",
"Weierstrass Extreme Value Theorem",
"Complex Modulus is Non-Negative",
"Complex Modulus equals Zero iff Zero",
"Weierstrass Extreme Value Theorem",
"Weierstrass Extreme Value Theorem",
"Weierstrass Extreme Value Theorem"
] |
proofwiki-17461 | Union with Disjoint Singleton is Dependent if Element Depends on Subset | Let $M = \struct{S, \mathscr I}$ be a matroid.
Let $A \subseteq S$.
Let $x \in S : x \notin A$.
Let $x$ depend on $A$.
Then $A \cup \set x$ is a dependent subset of $M$. | We proceed by Proof by Contraposition.
Let $A \cup \set x$ be independent.
By matroid axiom $(\text I 2)$:
:$A$ is independent
We have:
{{begin-eqn}}
{{eqn | l = \map \rho {A \cup \set x}
| r = \size {A \cup \set x}
| c = Rank of Independent Subset Equals Cardinality
}}
{{eqn | r = \size A + \size {\set x}... | Let $M = \struct{S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $A \subseteq S$.
Let $x \in S : x \notin A$.
Let $x$ [[Definition:Depends Relation (Matroid)|depend]] on $A$.
Then $A \cup \set x$ is a [[Definition:Dependent Subset (Matroid)|dependent subset]] of $M$. | We proceed by [[Proof by Contraposition]].
Let $A \cup \set x$ be [[Definition:Independent Subset (Matroid)|independent]].
By [[Axiom:Matroid Axioms|matroid axiom $(\text I 2)$]]:
:$A$ is [[Definition:Independent Subset (Matroid)|independent]]
We have:
{{begin-eqn}}
{{eqn | l = \map \rho {A \cup \set x}
| r... | Union with Disjoint Singleton is Dependent if Element Depends on Subset | https://proofwiki.org/wiki/Union_with_Disjoint_Singleton_is_Dependent_if_Element_Depends_on_Subset | https://proofwiki.org/wiki/Union_with_Disjoint_Singleton_is_Dependent_if_Element_Depends_on_Subset | [
"Matroid Dependence",
"Matroid Dependent Subsets"
] | [
"Definition:Matroid",
"Definition:Depends Relation (Matroid)",
"Definition:Matroid/Dependent Set"
] | [
"Proof by Contraposition",
"Definition:Matroid/Independent Set",
"Axiom:Matroid Axioms",
"Definition:Matroid/Independent Set",
"Rank of Independent Subset Equals Cardinality",
"Cardinality of Singleton",
"Rank of Independent Subset Equals Cardinality",
"Definition:Depends Relation (Matroid)",
"Rule ... |
proofwiki-17462 | Element Depends on Independent Set iff Union with Singleton is Dependent | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $X \in \mathscr I$.
Let $x \in S : x \notin X$.
Then:
:$x \in \map \sigma X$ {{iff}} $X \cup \set x$ is dependent. | === Necessary Condition ===
Let $x \in \map \sigma X$.
By definition of the closure:
:$x$ depends on $X$.
From Union with Disjoint Singleton is Dependent if Element Depends on Subset:
:$X \cup \set x$ is dependent.
{{qed|lemma}} | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $X \in \mathscr I$.
Let $x \in S : x \notin X$.
Then:
:$x \in \map \sigma X$ {{iff}} $X \cup \set x$ is [[Definition:Dependent Subset (Matroid)|dependent]]. | === Necessary Condition ===
Let $x \in \map \sigma X$.
By definition of the [[Definition:Closure Operator (Matroid)|closure]]:
:$x$ [[Definition:Depends Relation (Matroid)|depends]] on $X$.
From [[Union with Disjoint Singleton is Dependent if Element Depends on Subset]]:
:$X \cup \set x$ is [[Definition:Dependent Su... | Element Depends on Independent Set iff Union with Singleton is Dependent | https://proofwiki.org/wiki/Element_Depends_on_Independent_Set_iff_Union_with_Singleton_is_Dependent | https://proofwiki.org/wiki/Element_Depends_on_Independent_Set_iff_Union_with_Singleton_is_Dependent | [
"Matroid Dependence",
"Matroid Dependent Subsets",
"Element Depends on Independent Set iff Union with Singleton is Dependent"
] | [
"Definition:Matroid",
"Definition:Matroid/Dependent Set"
] | [
"Definition:Closure Operator (Matroid)",
"Definition:Depends Relation (Matroid)",
"Union with Disjoint Singleton is Dependent if Element Depends on Subset",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Dependent Set",
"Definition:Depends Relation (Matroid)"
] |
proofwiki-17463 | Element Depends on Independent Set iff Union with Singleton is Dependent/Lemma | Let $A \in \mathscr I$ such that $A \subseteq X \cup \set x$.
Then:
:$\size A \le \size X$ | ==== Case 1: $x \in A$ ====
Let $x \in A$.
We have:
{{begin-eqn}}
{{eqn | l = A \setminus \set x
| o = \subseteq
| r = \paren {X \cup \set x} \setminus \set x
| c = Set Difference over Subset
}}
{{eqn | r = \paren {X \setminus \set x} \cup \paren {\set x \setminus \set x}
| c = Set Difference is... | Let $A \in \mathscr I$ such that $A \subseteq X \cup \set x$.
Then:
:$\size A \le \size X$ | ==== Case 1: $x \in A$ ====
Let $x \in A$.
We have:
{{begin-eqn}}
{{eqn | l = A \setminus \set x
| o = \subseteq
| r = \paren {X \cup \set x} \setminus \set x
| c = [[Set Difference over Subset]]
}}
{{eqn | r = \paren {X \setminus \set x} \cup \paren {\set x \setminus \set x}
| c = [[Set Diffe... | Element Depends on Independent Set iff Union with Singleton is Dependent/Lemma | https://proofwiki.org/wiki/Element_Depends_on_Independent_Set_iff_Union_with_Singleton_is_Dependent/Lemma | https://proofwiki.org/wiki/Element_Depends_on_Independent_Set_iff_Union_with_Singleton_is_Dependent/Lemma | [
"Element Depends on Independent Set iff Union with Singleton is Dependent"
] | [] | [
"Set Difference over Subset",
"Set Difference is Right Distributive over Union",
"Set Difference with Disjoint Set",
"Set Difference with Superset is Empty Set",
"Union with Empty Set",
"Set Difference Union Second Set is Union",
"Definition:Matroid/Independent Set",
"Definition:Contradiction",
"Def... |
proofwiki-17464 | Rank of Independent Subset Equals Cardinality | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\rho : \powerset S \to \Z$ be the rank function of $M$.
Let $X \in \mathscr I$
Then:
:$\map \rho X = \size X$ | {{begin-eqn}}
{{eqn | l = \map \rho X
| r = \max \set {\size Y : Y \subseteq X \land X \in \mathscr I}
| c = {{Defof|Rank Function (Matroid)}}
}}
{{eqn | r = \max \set {\size Y : Y \in \powerset X \land X \in \mathscr I}
| c = {{Defof|Power Set}} of $X$
}}
{{eqn | r = \max \set {\size Y : Y \in \power... | 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 $X \in \mathscr I$
Then:
:$\map \rho X = \size X$ | {{begin-eqn}}
{{eqn | l = \map \rho X
| r = \max \set {\size Y : Y \subseteq X \land X \in \mathscr I}
| c = {{Defof|Rank Function (Matroid)}}
}}
{{eqn | r = \max \set {\size Y : Y \in \powerset X \land X \in \mathscr I}
| c = {{Defof|Power Set}} of $X$
}}
{{eqn | r = \max \set {\size Y : Y \in \power... | Rank of Independent Subset Equals Cardinality | https://proofwiki.org/wiki/Rank_of_Independent_Subset_Equals_Cardinality | https://proofwiki.org/wiki/Rank_of_Independent_Subset_Equals_Cardinality | [
"Matroid Rank Functions",
"Matroid Independent Subsets"
] | [
"Definition:Matroid",
"Definition:Rank Function (Matroid)"
] | [
"Axiom:Matroid Axioms",
"Cardinality of Proper Subset of Finite Set",
"Category:Matroid Rank Functions",
"Category:Matroid Independent Subsets"
] |
proofwiki-17465 | Generating Function for Lucas Numbers | Let $\map G z$ be the function defined as:
:$\map G z = \dfrac {2 - z} {1 - z - z^2}$
Then $\map G z$ is a generating function for the Lucas numbers. | Let the form of $\map G z$ be assumed as:
{{begin-eqn}}
{{eqn | l = \map G z
| r = \sum_{k \mathop \ge 0} L_k z^k
| c =
}}
{{eqn | r = L_0 + L_1 z + L_2 z^2 + L_3 z^3 + L_4 z^4 + \cdots
| c =
}}
{{eqn | r = 2 + z + 3 z^2 + 4 z^3 + 7 z^4 + \cdots
| c =
}}
{{end-eqn}}
where $L_n$ denotes the $n... | Let $\map G z$ be the [[Definition:Real Function|function]] defined as:
:$\map G z = \dfrac {2 - z} {1 - z - z^2}$
Then $\map G z$ is a [[Definition:Generating Function|generating function]] for the [[Definition:Lucas Number|Lucas numbers]]. | Let the form of $\map G z$ be assumed as:
{{begin-eqn}}
{{eqn | l = \map G z
| r = \sum_{k \mathop \ge 0} L_k z^k
| c =
}}
{{eqn | r = L_0 + L_1 z + L_2 z^2 + L_3 z^3 + L_4 z^4 + \cdots
| c =
}}
{{eqn | r = 2 + z + 3 z^2 + 4 z^3 + 7 z^4 + \cdots
| c =
}}
{{end-eqn}}
where $L_n$ denotes the ... | Generating Function for Lucas Numbers | https://proofwiki.org/wiki/Generating_Function_for_Lucas_Numbers | https://proofwiki.org/wiki/Generating_Function_for_Lucas_Numbers | [
"Lucas Numbers",
"Examples of Generating Functions"
] | [
"Definition:Real Function",
"Definition:Generating Function",
"Definition:Lucas Number"
] | [
"Definition:Lucas Number",
"Category:Lucas Numbers",
"Category:Examples of Generating Functions"
] |
proofwiki-17466 | 492 Cubed is Sum of 3 Positive Cubes in 13 Ways | The cube of $492$ can be expressed as the sum of $3$ positive cubes in $13$ different ways:
{{begin-eqn}}
{{eqn | l = 492^3
| r = 24^3 + 204^3 + 480^3
}}
{{eqn | r = 48^3 + 85^3 + 491^3
}}
{{eqn | r = 72^3 + 384^3 + 396^3
}}
{{eqn | r = 113^3 + 264^3 + 463^3
}}
{{eqn | r = 114^3 + 360^3 + 414^3
}}
{{eqn | r = 149... | Brute force. | The [[Definition:Cube (Algebra)|cube]] of $492$ can be expressed as the [[Definition:Integer Addition|sum]] of $3$ [[Definition:Positive Integer|positive]] [[Definition:Cube Number|cubes]] in $13$ different ways:
{{begin-eqn}}
{{eqn | l = 492^3
| r = 24^3 + 204^3 + 480^3
}}
{{eqn | r = 48^3 + 85^3 + 491^3
}}
{{e... | Brute force. | 492 Cubed is Sum of 3 Positive Cubes in 13 Ways | https://proofwiki.org/wiki/492_Cubed_is_Sum_of_3_Positive_Cubes_in_13_Ways | https://proofwiki.org/wiki/492_Cubed_is_Sum_of_3_Positive_Cubes_in_13_Ways | [
"492",
"Sums of Cubes"
] | [
"Definition:Cube/Algebra",
"Definition:Addition/Integers",
"Definition:Positive/Integer",
"Definition:Cube Number"
] | [] |
proofwiki-17467 | Largest Number not Expressible as Sum of Multiples of Coprime Integers/Generalization | Let $a, b$ be integers greater than $1$.
Let $d = \gcd \set {a, b}$.
Then the largest multiple of $d$ not expressible as a sum of multiples of $a$ and $b$ (possibly zero) is the number:
:$\dfrac {a b} d - a - b$ | By Integers Divided by GCD are Coprime:
:$\dfrac a d \perp \dfrac b d$
By Largest Number not Expressible as Sum of Multiples of Coprime Integers, the largest number not expressible as a sum of multiples of $\dfrac a d$ and $\dfrac b d$ is the number:
:$\dfrac {a b} {d^2} - \dfrac a d - \dfrac b d$
Let $k d$ be a multip... | Let $a, b$ be [[Definition:Integer|integers]] greater than $1$.
Let $d = \gcd \set {a, b}$.
Then the largest multiple of $d$ not expressible as a sum of multiples of $a$ and $b$ (possibly zero) is the number:
:$\dfrac {a b} d - a - b$ | By [[Integers Divided by GCD are Coprime]]:
:$\dfrac a d \perp \dfrac b d$
By [[Largest Number not Expressible as Sum of Multiples of Coprime Integers]], the largest number not expressible as a sum of multiples of $\dfrac a d$ and $\dfrac b d$ is the number:
:$\dfrac {a b} {d^2} - \dfrac a d - \dfrac b d$
Let $k d$ ... | Largest Number not Expressible as Sum of Multiples of Coprime Integers/Generalization | https://proofwiki.org/wiki/Largest_Number_not_Expressible_as_Sum_of_Multiples_of_Coprime_Integers/Generalization | https://proofwiki.org/wiki/Largest_Number_not_Expressible_as_Sum_of_Multiples_of_Coprime_Integers/Generalization | [
"Largest Number not Expressible as Sum of Multiples of Coprime Integers",
"Integer Combinations"
] | [
"Definition:Integer"
] | [
"Integers Divided by GCD are Coprime",
"Largest Number not Expressible as Sum of Multiples of Coprime Integers",
"Category:Largest Number not Expressible as Sum of Multiples of Coprime Integers",
"Category:Integer Combinations"
] |
proofwiki-17468 | Maximum Area of Isosceles Triangle | Consider two line segments $A$ and $B$ of equal length $a$ which are required to be the legs of an isosceles triangle $T$.
Then the area of $T$ is greatest when the apex of $T$ is a right angle.
The area of $T$ in this situation is equal to $\dfrac {a^2} 2$. | :500px
Let $\triangle OAB$ be the isosceles triangle $T$ formed by the legs $OA$ and $OB$.
Thus the apex of $T$ is at $O$.
Let $\theta$ be the angle $\angle AOB$.
We see that by keeping $OA$ fixed, $B$ can range over the semicircle $AOB$.
Thus $\theta$ can range from $0$ to $180 \degrees$, that is, $2$ right angles.
Fr... | Consider two [[Definition:Line Segment|line segments]] $A$ and $B$ of equal [[Definition:Length of Line|length]] $a$ which are required to be the [[Definition:Legs of Isosceles Triangle|legs]] of an [[Definition:Isosceles Triangle|isosceles triangle]] $T$.
Then the [[Definition:Area|area]] of $T$ is greatest when the ... | :[[File:Maximum-size-isosceles-triangle.png|500px]]
Let $\triangle OAB$ be the [[Definition:Isosceles Triangle|isosceles triangle]] $T$ formed by the [[Definition:Legs of Isosceles Triangle|legs]] $OA$ and $OB$.
Thus the [[Definition:Apex of Isosceles Triangle|apex]] of $T$ is at $O$.
Let $\theta$ be the [[Definitio... | Maximum Area of Isosceles Triangle | https://proofwiki.org/wiki/Maximum_Area_of_Isosceles_Triangle | https://proofwiki.org/wiki/Maximum_Area_of_Isosceles_Triangle | [
"Isosceles Triangles"
] | [
"Definition:Line/Segment",
"Definition:Linear Measure/Length",
"Definition:Triangle (Geometry)/Isosceles/Legs",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Area",
"Definition:Triangle (Geometry)/Isosceles/Apex",
"Definition:Right Angle",
"Definition:Area"
] | [
"File:Maximum-size-isosceles-triangle.png",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Triangle (Geometry)/Isosceles/Legs",
"Definition:Triangle (Geometry)/Isosceles/Apex",
"Definition:Angle",
"Definition:Circle/Semicircle",
"Definition:Right Angle",
"Area of Triangle in Terms of Two Side... |
proofwiki-17469 | Construction of Perpendicular using Rusty Compass | Let $AB$ be a line segment.
Using a straightedge and rusty compass, it is possible to construct a straight line at right angles to $AB$ from the endpoint $A$, without extending $AB$ past $A$. | As $DE = CD = DA$, the points $A$, $C$ and $E$ all lie on a circle of radius $AC$.
$CE$ is a straight line through the centers of circle $ACE$ and so is a diameter of circle $ACE$.
Hence by Thales' Theorem, $\angle CAE$ is a right angle
{{qed}} | Let $AB$ be a [[Definition:Line Segment|line segment]].
Using a [[Definition:Straightedge|straightedge]] and [[Definition:Rusty Compass|rusty compass]], it is possible to construct a [[Definition:Straight Line|straight line]] at [[Definition:Right Angle|right angles]] to $AB$ from the [[Definition:Endpoint of Line|end... | As $DE = CD = DA$, the [[Definition:Point|points]] $A$, $C$ and $E$ all lie on a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $AC$.
$CE$ is a [[Definition:Straight Line|straight line]] through the [[Definition:Center of Circle|centers]] of [[Definition:Circle|circle]] $ACE$ and so is a [[Defi... | Construction of Perpendicular using Rusty Compass | https://proofwiki.org/wiki/Construction_of_Perpendicular_using_Rusty_Compass | https://proofwiki.org/wiki/Construction_of_Perpendicular_using_Rusty_Compass | [
"Lines",
"Angles",
"Rusty Compass Constructions"
] | [
"Definition:Line/Segment",
"Definition:Straightedge",
"Definition:Rusty Compass",
"Definition:Line/Straight Line",
"Definition:Right Angle",
"Definition:Line/Endpoint"
] | [
"Definition:Point",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Line/Straight Line",
"Definition:Circle/Center",
"Definition:Circle",
"Definition:Diameter",
"Definition:Circle",
"Thales' Theorem",
"Definition:Right Angle"
] |
proofwiki-17470 | Sum to Infinity of 2x^2n over n by 2n Choose n | For $\cmod x < 1$:
:$\ds \frac {2 x \arcsin x} {\sqrt {1 - x^2} } = \sum_{n \mathop = 1}^\infty \frac {\paren {2 x}^{2 n} } {n \dbinom {2 n} n}$ | By Gregory Series:
:$\ds \arctan t = \sum_{m \mathop = 0}^\infty \frac {\paren {-1}^m t^{2 m + 1} } {2 m + 1}$
Let $t = \dfrac x {\sqrt {1 - x^2} }$.
Let $y = \arcsin x$.
Then:
{{begin-eqn}}
{{eqn | l = t
| r = \frac {\sin y} {\sqrt {1 - \sin^2 y} }
}}
{{eqn | r = \frac {\sin y} {\cos y}
| c = Sum of Square... | For $\cmod x < 1$:
:$\ds \frac {2 x \arcsin x} {\sqrt {1 - x^2} } = \sum_{n \mathop = 1}^\infty \frac {\paren {2 x}^{2 n} } {n \dbinom {2 n} n}$ | By [[Gregory Series]]:
:$\ds \arctan t = \sum_{m \mathop = 0}^\infty \frac {\paren {-1}^m t^{2 m + 1} } {2 m + 1}$
Let $t = \dfrac x {\sqrt {1 - x^2} }$.
Let $y = \arcsin x$.
Then:
{{begin-eqn}}
{{eqn | l = t
| r = \frac {\sin y} {\sqrt {1 - \sin^2 y} }
}}
{{eqn | r = \frac {\sin y} {\cos y}
| c = [[Sum... | Sum to Infinity of 2x^2n over n by 2n Choose n | https://proofwiki.org/wiki/Sum_to_Infinity_of_2x^2n_over_n_by_2n_Choose_n | https://proofwiki.org/wiki/Sum_to_Infinity_of_2x^2n_over_n_by_2n_Choose_n | [
"Central Binomial Coefficients"
] | [] | [
"Gregory Series",
"Sum of Squares of Sine and Cosine",
"Gregory Series",
"Translation of Index Variable of Summation",
"Binomial Theorem for Negative Index and Negative Parameter",
"Translation of Index Variable of Summation",
"Binomial Theorem",
"Definite Integral from 0 to Half Pi of Odd Power of Si... |
proofwiki-17471 | Definite Integral from -a to a of Power of a plus x by Power of a minus x | :$\ds \int_{-a}^a \paren {a + x}^{m - 1} \paren {a - x}^{n - 1} \rd x = \paren {2 a}^{m + n - 1} \frac {\map \Gamma m \map \Gamma n} {\map \Gamma {m + n} }$ | Note the resemblance of this result to the integral defining the beta function.
In view of this, we apply the substitution:
:$u = \dfrac {a + x} {2 a}$
We then have, by Derivative of Power:
:$\dfrac {\d u} {\d x} = \dfrac 1 {2 a}$
and:
{{begin-eqn}}
{{eqn | l = 1 - u
| r = 1 - \frac {a + x} {2 a}
}}
{{eqn | r =... | :$\ds \int_{-a}^a \paren {a + x}^{m - 1} \paren {a - x}^{n - 1} \rd x = \paren {2 a}^{m + n - 1} \frac {\map \Gamma m \map \Gamma n} {\map \Gamma {m + n} }$ | Note the resemblance of this result to the integral defining the [[Definition:Beta Function|beta function]].
In view of this, we apply the substitution:
:$u = \dfrac {a + x} {2 a}$
We then have, by [[Derivative of Power]]:
:$\dfrac {\d u} {\d x} = \dfrac 1 {2 a}$
and:
{{begin-eqn}}
{{eqn | l = 1 - u
| r =... | Definite Integral from -a to a of Power of a plus x by Power of a minus x | https://proofwiki.org/wiki/Definite_Integral_from_-a_to_a_of_Power_of_a_plus_x_by_Power_of_a_minus_x | https://proofwiki.org/wiki/Definite_Integral_from_-a_to_a_of_Power_of_a_plus_x_by_Power_of_a_minus_x | [
"Examples of Definite Integrals"
] | [] | [
"Definition:Beta Function",
"Power Rule for Derivatives",
"Integration by Substitution"
] |
proofwiki-17472 | Definite Integral from 0 to Pi of Logarithm of a plus b Cosine x | :$\ds \int_0^\pi \map \ln {a + b \cos x} \rd x = \pi \map \ln {\frac {a + \sqrt {a^2 - b^2} } 2}$ | Fix $b \in \R$ and define:
:$\ds \map I a = \int_0^\pi \map \ln {a + b \cos x} \rd x$
for $a \ge \size b$.
We have:
{{begin-eqn}}
{{eqn | l = \map {I'} a
| r = \frac \d {\d a} \int_0^\pi \map \ln {a + b \cos x} \rd x
}}
{{eqn | r = \int_0^\pi \frac \partial {\partial a} \paren {\map \ln {a + b \cos x} } \rd x
... | :$\ds \int_0^\pi \map \ln {a + b \cos x} \rd x = \pi \map \ln {\frac {a + \sqrt {a^2 - b^2} } 2}$ | Fix $b \in \R$ and define:
:$\ds \map I a = \int_0^\pi \map \ln {a + b \cos x} \rd x$
for $a \ge \size b$.
We have:
{{begin-eqn}}
{{eqn | l = \map {I'} a
| r = \frac \d {\d a} \int_0^\pi \map \ln {a + b \cos x} \rd x
}}
{{eqn | r = \int_0^\pi \frac \partial {\partial a} \paren {\map \ln {a + b \cos x} } \rd... | Definite Integral from 0 to Pi of Logarithm of a plus b Cosine x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_Logarithm_of_a_plus_b_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_Logarithm_of_a_plus_b_Cosine_x | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Cosine Function"
] | [] | [
"Definite Integral of Partial Derivative",
"Derivative of Natural Logarithm Function",
"Definite Integral of Even Function",
"Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Cosine x",
"Primitive of Root of x squared minus a squared/Logarithm Form",
"Primitive of Constant",
"Sum of Logarithms... |
proofwiki-17473 | Arccosine in terms of Arctangent | :$\arccos x = 2 \map \arctan {\sqrt {\dfrac {1 - x} {1 + x} } }$ | Let:
:$\theta = \arccos x$
Then by the definition of arccosine:
:$x = \cos \theta$
and:
:$0 \le \theta < \pi$
Then:
{{begin-eqn}}
{{eqn | l = 2 \map \arctan {\sqrt {\frac {1 - x} {1 + x} } }
| r = 2 \map \arctan {\sqrt {\frac {1 - \cos \theta} {1 + \cos \theta} } }
}}
{{eqn | r = 2 \map \arctan {\sqrt {\frac {... | :$\arccos x = 2 \map \arctan {\sqrt {\dfrac {1 - x} {1 + x} } }$ | Let:
:$\theta = \arccos x$
Then by the definition of [[Definition:Real Arccosine|arccosine]]:
:$x = \cos \theta$
and:
:$0 \le \theta < \pi$
Then:
{{begin-eqn}}
{{eqn | l = 2 \map \arctan {\sqrt {\frac {1 - x} {1 + x} } }
| r = 2 \map \arctan {\sqrt {\frac {1 - \cos \theta} {1 + \cos \theta} } }
}}
{{eqn |... | Arccosine in terms of Arctangent | https://proofwiki.org/wiki/Arccosine_in_terms_of_Arctangent | https://proofwiki.org/wiki/Arccosine_in_terms_of_Arctangent | [
"Arccosine Function",
"Arctangent Function"
] | [] | [
"Definition:Inverse Cosine/Real/Arccosine"
] |
proofwiki-17474 | Definite Integral from 0 to Half Pi of Reciprocal of a plus b Cosine x | :$\ds \int_0^{\pi/2} \frac 1 {a + b \cos x} \rd x = \frac 1 {\sqrt {a^2 - b^2} } \map \arccos {\frac b a}$ | Since $a > b > 0$, we have $a^2 > b^2$.
So:
{{begin-eqn}}
{{eqn | l = \int_0^{\pi/2} \frac 1 {a + b \cos x} \rd x
| r = \intlimits {\frac 2 {\sqrt {a^2 - b^2} } \map \arctan {\sqrt {\frac {a - b} {a + b} } \tan \frac x 2} } 0 1
| c = Primitive of $\dfrac 1 {p + q \cos x}$
}}
{{eqn | r = \frac 1 {\sqrt {a^2 - b^2} } \... | :$\ds \int_0^{\pi/2} \frac 1 {a + b \cos x} \rd x = \frac 1 {\sqrt {a^2 - b^2} } \map \arccos {\frac b a}$ | Since $a > b > 0$, we have $a^2 > b^2$.
So:
{{begin-eqn}}
{{eqn | l = \int_0^{\pi/2} \frac 1 {a + b \cos x} \rd x
| r = \intlimits {\frac 2 {\sqrt {a^2 - b^2} } \map \arctan {\sqrt {\frac {a - b} {a + b} } \tan \frac x 2} } 0 1
| c = [[Primitive of Reciprocal of p plus q by Cosine of a x|Primitive of $\dfrac 1 {p +... | Definite Integral from 0 to Half Pi of Reciprocal of a plus b Cosine x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Reciprocal_of_a_plus_b_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Reciprocal_of_a_plus_b_Cosine_x | [
"Definite Integrals involving Cosine Function"
] | [] | [
"Primitive of Reciprocal of p plus q by Cosine of a x",
"Arccosine in terms of Arctangent"
] |
proofwiki-17475 | Definite Integral to Infinity of Exponential of -a x by Sine of b x over x | :$\ds \int_0^\infty \frac {e^{-a x} \sin b x} x \rd x = \map \arctan {\frac b a}$ | Take $a$ constant and define:
:$\ds \map I b = \int_0^\infty \frac {e^{-a x} \sin b x} x \rd x$
We have:
{{begin-eqn}}
{{eqn | l = \map {I'} b
| r = \frac \d {\d b} \int_0^\infty \frac {e^{-a x} \sin b x} x \rd x
}}
{{eqn | r = \int_0^\infty \frac \partial {\partial b} \paren {\frac {e^{-a x} \sin b x} x} \rd x
| c... | :$\ds \int_0^\infty \frac {e^{-a x} \sin b x} x \rd x = \map \arctan {\frac b a}$ | Take $a$ constant and define:
:$\ds \map I b = \int_0^\infty \frac {e^{-a x} \sin b x} x \rd x$
We have:
{{begin-eqn}}
{{eqn | l = \map {I'} b
| r = \frac \d {\d b} \int_0^\infty \frac {e^{-a x} \sin b x} x \rd x
}}
{{eqn | r = \int_0^\infty \frac \partial {\partial b} \paren {\frac {e^{-a x} \sin b x} x} \rd x
... | Definite Integral to Infinity of Exponential of -a x by Sine of b x over x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_by_Sine_of_b_x_over_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_by_Sine_of_b_x_over_x | [
"Definite Integrals involving Sine Function",
"Definite Integrals involving Exponential Function"
] | [] | [
"Definite Integral of Partial Derivative",
"Derivative of Cosine Function/Corollary",
"Definite Integral to Infinity of Exponential of -a x by Cosine of b x",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-17476 | Definite Integral to Infinity of Sine of m x over Exponential of 2 Pi x minus One | :$\ds \int_0^\infty \frac {\sin m x} {e^{2 \pi x} - 1} \rd x = \frac 1 4 \coth \frac m 2 - \frac 1 {2 m}$ | We have:
{{begin-eqn}}
{{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sin m x} {e^{2 \pi x} - 1} \rd x
| r = \int_0^\infty \frac {e^{-2 \pi x} \sin m x} {1 - e^{-2 \pi x} } \rd x
}}
{{eqn | r = \int_0^\infty e^{-2 \pi x} \sin m x \paren {\sum_{k = 0}^\infty e^{-2 \pi k x} } \rd x
| c = Sum of Infinite Geometric Sequ... | :$\ds \int_0^\infty \frac {\sin m x} {e^{2 \pi x} - 1} \rd x = \frac 1 4 \coth \frac m 2 - \frac 1 {2 m}$ | We have:
{{begin-eqn}}
{{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sin m x} {e^{2 \pi x} - 1} \rd x
| r = \int_0^\infty \frac {e^{-2 \pi x} \sin m x} {1 - e^{-2 \pi x} } \rd x
}}
{{eqn | r = \int_0^\infty e^{-2 \pi x} \sin m x \paren {\sum_{k = 0}^\infty e^{-2 \pi k x} } \rd x
| c = [[Sum of Infinite Geometric S... | Definite Integral to Infinity of Sine of m x over Exponential of 2 Pi x minus One | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Sine_of_m_x_over_Exponential_of_2_Pi_x_minus_One | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Sine_of_m_x_over_Exponential_of_2_Pi_x_minus_One | [
"Definite Integrals involving Sine Function",
"Definite Integrals involving Exponential Function"
] | [] | [
"Sum of Infinite Geometric Sequence",
"Definite Integral to Infinity of Exponential of -a x by Sine of b x",
"Mittag-Leffler Expansion for Hyperbolic Cotangent Function",
"Definition:Integer"
] |
proofwiki-17477 | Definite Integral to Infinity of Exponential of -a x^2 by Cosine of b x | :$\ds \int_0^\infty e^{-a x^2} \cos b x \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-\frac {b^2} {4 a} }$ | Fix $a$ and define:
:$\ds \map I b = \int_0^\infty e^{-a x^2} \cos b x \rd x$
for all $b \in \R$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \map {I'} b
| r = \frac \d {\d b} \paren {\int_0^\infty e^{-a x^2} \cos b x \rd x}
}}
{{eqn | r = \int_0^\infty \frac \partial {\partial b} \paren {e^{-a x^2} \cos b x} \rd x
|... | :$\ds \int_0^\infty e^{-a x^2} \cos b x \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-\frac {b^2} {4 a} }$ | Fix $a$ and define:
:$\ds \map I b = \int_0^\infty e^{-a x^2} \cos b x \rd x$
for all $b \in \R$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \map {I'} b
| r = \frac \d {\d b} \paren {\int_0^\infty e^{-a x^2} \cos b x \rd x}
}}
{{eqn | r = \int_0^\infty \frac \partial {\partial b} \paren {e^{-a x^2} \cos b x} \rd ... | Definite Integral to Infinity of Exponential of -a x^2 by Cosine of b x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x^2_by_Cosine_of_b_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x^2_by_Cosine_of_b_x | [
"Definite Integrals involving Exponential Function",
"Definite Integrals involving Cosine Function"
] | [] | [
"Definite Integral of Partial Derivative",
"Derivative of Cosine Function/Corollary",
"Integration by Parts",
"Exponential Tends to Zero and Infinity",
"Primitive of Function under its Derivative",
"Primitive of Constant",
"Definite Integral to Infinity of Exponential of -a x^2",
"Exponential of Zero"... |
proofwiki-17478 | Fourier Series for Logarithm of Sine of x over 0 to Pi | :$\ds \map \ln {\sin x} = -\ln 2 - \sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n$
where $0 < x < \pi$. | We find the Half-Range Fourier Cosine Series over $\openint 0 {\dfrac \pi 2}$ for $\map \ln {\sin x}$.
By definition:
:$\ds \map \ln {\sin x} \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos 2 n x$
where for all $n \in \Z_{\ge 0}$:
:$\ds a_n = \frac 4 \pi \int_0^{\pi/2} \map \ln {\sin x} \cos 2 n x \ \d x$
By ... | :$\ds \map \ln {\sin x} = -\ln 2 - \sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n$
where $0 < x < \pi$. | We find the [[Definition:Half-Range Fourier Cosine Series|Half-Range Fourier Cosine Series]] over $\openint 0 {\dfrac \pi 2}$ for $\map \ln {\sin x}$.
By definition:
:$\ds \map \ln {\sin x} \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos 2 n x$
where for all $n \in \Z_{\ge 0}$:
:$\ds a_n = \frac 4 \pi \int_... | Fourier Series for Logarithm of Sine of x over 0 to Pi/Proof 1 | https://proofwiki.org/wiki/Fourier_Series_for_Logarithm_of_Sine_of_x_over_0_to_Pi | https://proofwiki.org/wiki/Fourier_Series_for_Logarithm_of_Sine_of_x_over_0_to_Pi/Proof_1 | [
"Examples of Fourier Series",
"Fourier Series for Logarithm of Sine of x over 0 to Pi"
] | [] | [
"Definition:Half-Range Fourier Cosine Series",
"Definite Integral from 0 to Half Pi of Logarithm of Sine x",
"Definite Integral from 0 to Half Pi of Logarithm of Sine x by Cosine of 2nx"
] |
proofwiki-17479 | Fourier Series for Logarithm of Sine of x over 0 to Pi | :$\ds \map \ln {\sin x} = -\ln 2 - \sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n$
where $0 < x < \pi$. | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty \dfrac {\cos 2 n x} n
| r = \sum_{n \mathop = 1}^\infty \dfrac {\map \exp {2 i n x} + \map \exp {-2 i n x} } {2 n}
| c = Euler's Cosine Identity: $\cos z = \dfrac {\map \exp {i z} + \map \exp {-i z} } 2$
}}
{{eqn | r = \frac 1 2 \sum_{n \mathop = 1}^\inf... | :$\ds \map \ln {\sin x} = -\ln 2 - \sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n$
where $0 < x < \pi$. | {{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty \dfrac {\cos 2 n x} n
| r = \sum_{n \mathop = 1}^\infty \dfrac {\map \exp {2 i n x} + \map \exp {-2 i n x} } {2 n}
| c = [[Euler's Cosine Identity]]: $\cos z = \dfrac {\map \exp {i z} + \map \exp {-i z} } 2$
}}
{{eqn | r = \frac 1 2 \sum_{n \mathop = 1}^... | Fourier Series for Logarithm of Sine of x over 0 to Pi/Proof 2 | https://proofwiki.org/wiki/Fourier_Series_for_Logarithm_of_Sine_of_x_over_0_to_Pi | https://proofwiki.org/wiki/Fourier_Series_for_Logarithm_of_Sine_of_x_over_0_to_Pi/Proof_2 | [
"Examples of Fourier Series",
"Fourier Series for Logarithm of Sine of x over 0 to Pi"
] | [] | [
"Euler's Cosine Identity",
"Exponent Combination Laws/Power of Power",
"Power Series Expansion for Logarithm of 1 - x",
"Sum of Logarithms",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Euler's Cosine Identity",
"Logarithm of Power/Natural Logarithm",
"Sum of Logarithms"
] |
proofwiki-17480 | Subset of Set Difference iff Disjoint Set | Let $S, T$ be sets.
Let $A \subseteq S$
Then:
:$A \cap T = \O \iff A \subseteq S \setminus T$
where:
:$A \cap T$ denotes set intersection
:$\O$ denotes the empty set
:$S \setminus T$ denotes set difference. | We have:
{{begin-eqn}}
{{eqn | l = A \cap \paren {S \setminus T}
| r = \paren {A \cap S} \setminus T
| c = Intersection with Set Difference is Set Difference with Intersection
}}
{{eqn | r = A \setminus T
| c = Intersection with Subset is Subset
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = A
... | Let $S, T$ be [[Definition:Set|sets]].
Let $A \subseteq S$
Then:
:$A \cap T = \O \iff A \subseteq S \setminus T$
where:
:$A \cap T$ denotes [[Definition:Set Intersection|set intersection]]
:$\O$ denotes the [[Definition:Empty Set|empty set]]
:$S \setminus T$ denotes [[Definition:Set Difference|set difference]]. | We have:
{{begin-eqn}}
{{eqn | l = A \cap \paren {S \setminus T}
| r = \paren {A \cap S} \setminus T
| c = [[Intersection with Set Difference is Set Difference with Intersection]]
}}
{{eqn | r = A \setminus T
| c = [[Intersection with Subset is Subset]]
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l =... | Subset of Set Difference iff Disjoint Set | https://proofwiki.org/wiki/Subset_of_Set_Difference_iff_Disjoint_Set | https://proofwiki.org/wiki/Subset_of_Set_Difference_iff_Disjoint_Set | [
"Set Difference",
"Disjoint Sets"
] | [
"Definition:Set",
"Definition:Set Intersection",
"Definition:Empty Set",
"Definition:Set Difference"
] | [
"Intersection with Set Difference is Set Difference with Intersection",
"Intersection with Subset is Subset",
"Intersection with Subset is Subset",
"Set Difference with Disjoint Set",
"Category:Set Difference",
"Category:Disjoint Sets"
] |
proofwiki-17481 | Set Difference of Doubleton and Singleton is Singleton | Let $x, y$ be distinct objects.
Then:
:$\set{x, y} \setminus \set x = \set y$ | {{begin-eqn}}
{{eqn | l = \set {x, y} \setminus \set x
| r = \set {z: z \in \set {x, y} \land z \notin \set x}
| c = {{Defof|Set Difference}}
}}
{{eqn | r = \set {z: \paren {z = x \lor z = y} \land z \notin \set x}
| c = {{Defof|Doubleton}}
}}
{{eqn | r = \set {z: \paren {z = x \lor z = y} \land z \ne... | Let $x, y$ be [[Definition:Distinct|distinct]] [[Definition:Object|objects]].
Then:
:$\set{x, y} \setminus \set x = \set y$ | {{begin-eqn}}
{{eqn | l = \set {x, y} \setminus \set x
| r = \set {z: z \in \set {x, y} \land z \notin \set x}
| c = {{Defof|Set Difference}}
}}
{{eqn | r = \set {z: \paren {z = x \lor z = y} \land z \notin \set x}
| c = {{Defof|Doubleton}}
}}
{{eqn | r = \set {z: \paren {z = x \lor z = y} \land z \ne... | Set Difference of Doubleton and Singleton is Singleton | https://proofwiki.org/wiki/Set_Difference_of_Doubleton_and_Singleton_is_Singleton | https://proofwiki.org/wiki/Set_Difference_of_Doubleton_and_Singleton_is_Singleton | [
"Singletons",
"Doubletons"
] | [
"Definition:Distinct",
"Definition:Object"
] | [
"Rule of Distribution/Conjunction Distributes over Disjunction",
"Disjunction with Contradiction",
"Rule of Simplification",
"Category:Singletons",
"Category:Doubletons"
] |
proofwiki-17482 | Egyptian Formula for Area of Quadrilateral | Let $\Box ABCD$ be a quadrilateral.
Let the sides of $\Box ABCD$ be $a$, $b$, $c$ and $d$ such that $a$ is opposite $c$ and $b$ is opposite $d$.
Then the area of $\Box ABCD$ can be approximated by:
:$\map \Area {\Box ABCD} \approx \dfrac {a + c} 2 \times \dfrac {b + d} 2$
The closer $\Box ABCD$ is to a rectangle, the b... | thumbA [[Definition:Quadrilateralquadrilateral $\Box ABCD$ with side lengths $a = \overline {AB}$, $b = \overline {BC}$, $c = \overline {CD}$, and $d = \overline {DA}$. The point $C'$ is positioned so that $\Box ABC'D$ is approximately a rectangle if $\angle DAB$ is approximately a right angle.]]
We have a quadrilatera... | Let $\Box ABCD$ be a [[Definition:Quadrilateral|quadrilateral]].
Let the [[Definition:Side of Polygon|sides]] of $\Box ABCD$ be $a$, $b$, $c$ and $d$ such that $a$ is [[Definition:Opposite (in Polygon)|opposite]] $c$ and $b$ is [[Definition:Opposite (in Polygon)|opposite]] $d$.
Then the [[Definition:Area|area]] of $... | [[File:Egyptian-quadrilateral-area-figure.png|thumb|A [[Definition:Quadrilateral|quadrilateral]] $\Box ABCD$ with [[Definition:Side of Polygon|side]] [[Definition:Length of Line|lengths]] $a = \overline {AB}$, $b = \overline {BC}$, $c = \overline {CD}$, and $d = \overline {DA}$. The [[Definition:Point|point]] $C'$ is p... | Egyptian Formula for Area of Quadrilateral/Proof 1 | https://proofwiki.org/wiki/Egyptian_Formula_for_Area_of_Quadrilateral | https://proofwiki.org/wiki/Egyptian_Formula_for_Area_of_Quadrilateral/Proof_1 | [
"Egyptian Formula for Area of Quadrilateral",
"Quadrilaterals",
"Area Formulas"
] | [
"Definition:Quadrilateral",
"Definition:Polygon/Side",
"Definition:Polygon/Opposite",
"Definition:Polygon/Opposite",
"Definition:Area",
"Definition:Approximation",
"Definition:Quadrilateral/Rectangle",
"Definition:Approximation"
] | [
"File:Egyptian-quadrilateral-area-figure.png",
"Definition:Quadrilateral",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Point",
"Definition:Approximation",
"Definition:Quadrilateral/Rectangle",
"Definition:Approximation",
"Definition:Right Angle",
"Definition:Quadrila... |
proofwiki-17483 | 1 plus Perfect Power is not Power of 2 | The equation:
:$1 + a^n = 2^m$
has no solutions in the integers for $n, m > 1$.
This is an elementary special case of Catalan's Conjecture. | {{AimForCont}} there is a solution.
Then:
{{begin-eqn}}
{{eqn | l = a^n
| r = 2^m - 1
}}
{{eqn | o = \equiv
| r = -1
| rr = \pmod 4
| c = as $m > 1$
}}
{{end-eqn}}
$a$ is immediately seen to be odd.
By Square Modulo 4, $n$ must also be odd.
Now:
{{begin-eqn}}
{{eqn | l = 2^m
| r = a^n + 1
... | The equation:
:$1 + a^n = 2^m$
has no solutions in the [[Definition:Integer|integers]] for $n, m > 1$.
This is an elementary special case of [[Catalan's Conjecture]]. | {{AimForCont}} there is a solution.
Then:
{{begin-eqn}}
{{eqn | l = a^n
| r = 2^m - 1
}}
{{eqn | o = \equiv
| r = -1
| rr = \pmod 4
| c = as $m > 1$
}}
{{end-eqn}}
$a$ is immediately seen to be [[Definition:Odd Integer|odd]].
By [[Square Modulo 4]], $n$ must also be [[Definition:Odd Integer|o... | 1 plus Perfect Power is not Power of 2 | https://proofwiki.org/wiki/1_plus_Perfect_Power_is_not_Power_of_2 | https://proofwiki.org/wiki/1_plus_Perfect_Power_is_not_Power_of_2 | [
"Number Theory"
] | [
"Definition:Integer",
"Catalan's Conjecture"
] | [
"Definition:Odd Integer",
"Square Modulo 4",
"Definition:Odd Integer",
"Sum of Two Odd Powers",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Contradiction",
"Proof by Contradiction",
"Category:Number Theory"
] |
proofwiki-17484 | Definite Integral from 0 to Half Pi of Logarithm of Sine x by Cosine of 2nx | For $n \in \N_{>0}$:
:$\ds \int_0^{\pi/2} \map \ln {\sin x} \cos 2 n x \rd x = -\frac \pi {4 n}$ | First we have:
{{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \map \ln {\sin x} \sin 2 n x
| r = \lim_{x \mathop \to 0} \frac {\map \ln {\sin x} } {\csc 2 n x}
| c = {{Defof|Cosecant}}
}}
{{eqn | r = \lim_{x \mathop \to 0} \frac {\cot x} {- 2 n \cot 2 n x \csc 2 n x}
| c = L'Hôpital's Rule: Corollar... | For $n \in \N_{>0}$:
:$\ds \int_0^{\pi/2} \map \ln {\sin x} \cos 2 n x \rd x = -\frac \pi {4 n}$ | First we have:
{{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to 0} \map \ln {\sin x} \sin 2 n x
| r = \lim_{x \mathop \to 0} \frac {\map \ln {\sin x} } {\csc 2 n x}
| c = {{Defof|Cosecant}}
}}
{{eqn | r = \lim_{x \mathop \to 0} \frac {\cot x} {- 2 n \cot 2 n x \csc 2 n x}
| c = [[L'Hôpital's Rule/Corolla... | Definite Integral from 0 to Half Pi of Logarithm of Sine x by Cosine of 2nx | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Logarithm_of_Sine_x_by_Cosine_of_2nx | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Logarithm_of_Sine_x_by_Cosine_of_2nx | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Sine Function"
] | [] | [
"L'Hôpital's Rule/Corollary 2",
"L'Hôpital's Rule",
"Primitive of Cosine Function/Corollary",
"Integration by Parts",
"Primitive of Cotangent Function",
"Werner Formulas/Sine by Cosine",
"Integration by Substitution",
"Lagrange's Trigonometric Identities/Cosine",
"Integral of Constant/Definite",
"... |
proofwiki-17485 | Distinct Matroid Elements are Parallel iff Each is in Closure of Other/Lemma | Let $a, b \in S$.
Let $\set a$ and $\set b$ be independent.
Then $\set {a, b}$ is dependent {{iff}}:
:$a \in \map \sigma {\set b}$
and
:$b \in \map \sigma {\set a}$ | {{begin-eqn}}
{{eqn | l = \set {a, b}
| o = \notin
| r = \mathscr I
}}
{{eqn | ll= \leadstoandfrom
| l = \set a \cup \set b
| o = \notin
| r = \mathscr I
| c = Union of Disjoint Singletons is Doubleton
}}
{{eqn | ll= \leadstoandfrom
| l = a
| o = \in
| r = \map \sig... | Let $a, b \in S$.
Let $\set a$ and $\set b$ be [[Definition:Independent Subset (Matroid)|independent]].
Then $\set {a, b}$ is [[Definition:Dependent Subset (Matroid)|dependent]] {{iff}}:
:$a \in \map \sigma {\set b}$
and
:$b \in \map \sigma {\set a}$ | {{begin-eqn}}
{{eqn | l = \set {a, b}
| o = \notin
| r = \mathscr I
}}
{{eqn | ll= \leadstoandfrom
| l = \set a \cup \set b
| o = \notin
| r = \mathscr I
| c = [[Union of Disjoint Singletons is Doubleton]]
}}
{{eqn | ll= \leadstoandfrom
| l = a
| o = \in
| r = \map ... | Distinct Matroid Elements are Parallel iff Each is in Closure of Other/Lemma | https://proofwiki.org/wiki/Distinct_Matroid_Elements_are_Parallel_iff_Each_is_in_Closure_of_Other/Lemma | https://proofwiki.org/wiki/Distinct_Matroid_Elements_are_Parallel_iff_Each_is_in_Closure_of_Other/Lemma | [
"Distinct Matroid Elements are Parallel iff Each is in Closure of Other"
] | [
"Definition:Matroid/Independent Set",
"Definition:Matroid/Dependent Set"
] | [
"Union of Disjoint Singletons is Doubleton",
"Element Depends on Independent Set iff Union with Singleton is Dependent",
"Element Depends on Independent Set iff Union with Singleton is Dependent",
"Category:Distinct Matroid Elements are Parallel iff Each is in Closure of Other"
] |
proofwiki-17486 | Definite Integral to Infinity of Sine m x over x by x Squared plus a Squared | :$\ds \int_0^\infty \frac {\sin m x} {x \paren {x^2 + a^2} } \rd x = \frac \pi {2 a^2} \paren {1 - e^{-m a} }$ | Fix $a$ and set:
:$\ds \map I m = \int_0^\infty \frac {\sin m x} {x \paren {x^2 + a^2} } \rd x$
for $m \ge 0$.
We have:
{{begin-eqn}}
{{eqn | l = \map {I'} m
| r = \frac \d {\d m} \int_0^\infty \frac {\sin m x} {x \paren {x^2 + a^2} } \rd x
}}
{{eqn | r = \int_0^\infty \frac \partial {\partial m} \paren {\frac {\sin... | :$\ds \int_0^\infty \frac {\sin m x} {x \paren {x^2 + a^2} } \rd x = \frac \pi {2 a^2} \paren {1 - e^{-m a} }$ | Fix $a$ and set:
:$\ds \map I m = \int_0^\infty \frac {\sin m x} {x \paren {x^2 + a^2} } \rd x$
for $m \ge 0$.
We have:
{{begin-eqn}}
{{eqn | l = \map {I'} m
| r = \frac \d {\d m} \int_0^\infty \frac {\sin m x} {x \paren {x^2 + a^2} } \rd x
}}
{{eqn | r = \int_0^\infty \frac \partial {\partial m} \paren {\frac {... | Definite Integral to Infinity of Sine m x over x by x Squared plus a Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Sine_m_x_over_x_by_x_Squared_plus_a_Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Sine_m_x_over_x_by_x_Squared_plus_a_Squared | [
"Definite Integrals involving Sine Function"
] | [] | [
"Definite Integral of Partial Derivative",
"Derivative of Sine Function/Corollary",
"Definite Integral to Infinity of Cosine m x over x Squared plus a Squared",
"Primitive of Exponential of a x",
"Sine of Zero is Zero",
"Exponential of Zero"
] |
proofwiki-17487 | Definite Integral from 0 to Pi of a Squared minus 2 a b Cosine x plus b Squared | :$\ds \int_0^\pi \map \ln {a^2 - 2 a b \cos x + b^2} \rd x = \begin{cases}2 \pi \ln a & a \ge b > 0 \\ 2 \pi \ln b & b \ge a > 0\end{cases}$ | Note that:
:$\paren {a - b}^2 \ge 0$
so by Square of Sum:
:$a^2 - 2 a b + b^2 \ge 0$
So:
:$a^2 + b^2 \ge 2 a b = \size {-2 a b}$
so we may apply Definite Integral from $0$ to $\pi$ of $\map \ln {a + b \cos x}$.
We then have:
{{begin-eqn}}
{{eqn | l = \int_0^\pi \map \ln {a^2 - 2 a b \cos x + b^2} \rd x
| r = \pi \map... | :$\ds \int_0^\pi \map \ln {a^2 - 2 a b \cos x + b^2} \rd x = \begin{cases}2 \pi \ln a & a \ge b > 0 \\ 2 \pi \ln b & b \ge a > 0\end{cases}$ | Note that:
:$\paren {a - b}^2 \ge 0$
so by [[Square of Sum]]:
:$a^2 - 2 a b + b^2 \ge 0$
So:
:$a^2 + b^2 \ge 2 a b = \size {-2 a b}$
so we may apply [[Definite Integral from 0 to Pi of Logarithm of a plus b Cosine x|Definite Integral from $0$ to $\pi$ of $\map \ln {a + b \cos x}$]].
We then have:
{{begin-eqn}}... | Definite Integral from 0 to Pi of a Squared minus 2 a b Cosine x plus b Squared | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_a_Squared_minus_2_a_b_Cosine_x_plus_b_Squared | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_a_Squared_minus_2_a_b_Cosine_x_plus_b_Squared | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Cosine Function"
] | [] | [
"Square of Sum",
"Definite Integral from 0 to Pi of Logarithm of a plus b Cosine x",
"Logarithm of Power",
"Logarithm of Power"
] |
proofwiki-17488 | Definite Integral from 0 to Half Pi of Square of Logarithm of Sine x | :$\ds \int_0^{\pi/2} \paren {\map \ln {\sin x} }^2 \rd x = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}$ | From Fourier Series for $\map \ln {\sin x}$ from $0$ to $\pi$:
:$\ds \map \ln {\sin x} = -\ln 2 - \sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n$
Then, by Parseval's Theorem:
{{begin-eqn}}
{{eqn | l = \frac 2 \pi \int_0^\pi \paren {\map \ln {\sin x} }^2 \rd x
| r = 2 \paren {\ln 2}^2 + \sum_{n = 1}^\infty \frac... | :$\ds \int_0^{\pi/2} \paren {\map \ln {\sin x} }^2 \rd x = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}$ | From [[Fourier Series for Logarithm of Sine of x over 0 to Pi|Fourier Series for $\map \ln {\sin x}$ from $0$ to $\pi$]]:
:$\ds \map \ln {\sin x} = -\ln 2 - \sum_{n \mathop = 1}^\infty \frac {\cos 2 n x} n$
Then, by [[Parseval's Theorem]]:
{{begin-eqn}}
{{eqn | l = \frac 2 \pi \int_0^\pi \paren {\map \ln {\sin x} }^... | Definite Integral from 0 to Half Pi of Square of Logarithm of Sine x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Square_of_Logarithm_of_Sine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Square_of_Logarithm_of_Sine_x | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Sine Function"
] | [] | [
"Fourier Series for Logarithm of Sine of x over 0 to Pi",
"Parseval's Theorem",
"Basel Problem",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Sine of Supplementary Angle"
] |
proofwiki-17489 | Definite Integral from 0 to Half Pi of Square of Logarithm of Cosine x | :$\ds \int_0^{\pi/2} \paren {\map \ln {\cos x} }^2 \rd x = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}$ | {{begin-eqn}}
{{eqn | l = \int_0^{\pi/2} \paren {\map \ln {\cos x} }^2 \rd x
| r = \int_0^{\pi/2} \paren {\map \ln {\map \cos {\frac \pi 2 - x} } }^2 \rd x
}}
{{eqn | r = \int_0^{\pi/2} \paren {\map \ln {\sin x} }^2 \rd x
| c = Cosine of Complement equals Sine
}}
{{eqn | r = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^... | :$\ds \int_0^{\pi/2} \paren {\map \ln {\cos x} }^2 \rd x = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}$ | {{begin-eqn}}
{{eqn | l = \int_0^{\pi/2} \paren {\map \ln {\cos x} }^2 \rd x
| r = \int_0^{\pi/2} \paren {\map \ln {\map \cos {\frac \pi 2 - x} } }^2 \rd x
}}
{{eqn | r = \int_0^{\pi/2} \paren {\map \ln {\sin x} }^2 \rd x
| c = [[Cosine of Complement equals Sine]]
}}
{{eqn | r = \frac \pi 2 \paren {\ln 2}^2 + \frac {... | Definite Integral from 0 to Half Pi of Square of Logarithm of Cosine x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Square_of_Logarithm_of_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Square_of_Logarithm_of_Cosine_x | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Cosine Function"
] | [] | [
"Cosine of Complement equals Sine",
"Definite Integral from 0 to Half Pi of Square of Logarithm of Sine x"
] |
proofwiki-17490 | Definite Integral from 0 to 2 Pi of Logarithm of a plus b Cosine x | :$\ds \int_0^{2 \pi} \map \ln {a + b \cos x} \rd x = 2 \pi \map \ln {\frac {a + \sqrt {a^2 - b^2} } 2}$ | {{begin-eqn}}
{{eqn | l = \int_0^{2 \pi} \map \ln {a + b \cos x} \rd x
| r = \int_0^\pi \map \ln {a + b \cos x} \rd x + \int_\pi^{2 \pi} \map \ln {a + b \cos x} \rd x
| c = Sum of Integrals on Adjacent Intervals for Integrable Functions
}}
{{eqn | r = \int_0^\pi \map \ln {a + b \cos x} \rd x - \int_\pi^0 \m... | :$\ds \int_0^{2 \pi} \map \ln {a + b \cos x} \rd x = 2 \pi \map \ln {\frac {a + \sqrt {a^2 - b^2} } 2}$ | {{begin-eqn}}
{{eqn | l = \int_0^{2 \pi} \map \ln {a + b \cos x} \rd x
| r = \int_0^\pi \map \ln {a + b \cos x} \rd x + \int_\pi^{2 \pi} \map \ln {a + b \cos x} \rd x
| c = [[Sum of Integrals on Adjacent Intervals for Integrable Functions]]
}}
{{eqn | r = \int_0^\pi \map \ln {a + b \cos x} \rd x - \int_\pi^... | Definite Integral from 0 to 2 Pi of Logarithm of a plus b Cosine x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_2_Pi_of_Logarithm_of_a_plus_b_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_2_Pi_of_Logarithm_of_a_plus_b_Cosine_x | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Cosine Function"
] | [] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Integration by Substitution",
"Reversal of Limits of Definite Integral",
"Cosine of Conjugate Angle",
"Definite Integral from 0 to Pi of Logarithm of a plus b Cosine x"
] |
proofwiki-17491 | Definite Integral from 0 to Pi of Sec x by Logarithm of One plus b Cosine x over One plus a Cosine x | :$\ds \int_0^{\pi/2} \sec x \map \ln {\frac {1 + b \cos x} {1 + a \cos x} } \rd x = \frac 1 2 \paren {\paren {\arccos a}^2 - \paren {\arccos b}^2}$ | Note that by Difference of Logarithms:
:$\ds \int_0^{\pi/2} \sec x \map \ln {\frac {1 + b \cos x} {1 + a \cos x} } \rd x = \int_0^{\pi/2} \sec x \map \ln {1 + b \cos x} \rd x - \int_0^{\pi/2} \sec \map \ln {1 + a \cos x} \rd x$
For each $\alpha \in \openint {-1} 1$, set:
:$\ds \map I \alpha = \int_0^{\pi/2} \sec x \m... | :$\ds \int_0^{\pi/2} \sec x \map \ln {\frac {1 + b \cos x} {1 + a \cos x} } \rd x = \frac 1 2 \paren {\paren {\arccos a}^2 - \paren {\arccos b}^2}$ | Note that by [[Difference of Logarithms]]:
:$\ds \int_0^{\pi/2} \sec x \map \ln {\frac {1 + b \cos x} {1 + a \cos x} } \rd x = \int_0^{\pi/2} \sec x \map \ln {1 + b \cos x} \rd x - \int_0^{\pi/2} \sec \map \ln {1 + a \cos x} \rd x$
For each $\alpha \in \openint {-1} 1$, set:
:$\ds \map I \alpha = \int_0^{\pi/2} \s... | Definite Integral from 0 to Pi of Sec x by Logarithm of One plus b Cosine x over One plus a Cosine x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_Sec_x_by_Logarithm_of_One_plus_b_Cosine_x_over_One_plus_a_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_Sec_x_by_Logarithm_of_One_plus_b_Cosine_x_over_One_plus_a_Cosine_x | [
"Definite Integrals involving Cosine Function",
"Definite Integrals involving Logarithm Function"
] | [] | [
"Difference of Logarithms",
"Definite Integral of Partial Derivative",
"Derivative of Composite Function",
"Derivative of Natural Logarithm Function",
"Definite Integral from 0 to Half Pi of Reciprocal of a plus b Cosine x",
"Fundamental Theorem of Calculus/Second Part",
"Derivative of Arccosine Functio... |
proofwiki-17492 | Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x by Secant of p x | :$\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \sec p x} \rd x = \frac 1 2 \map \ln {\frac {b^2 + p^2} {a^2 + p^2} }$ | Fix $p$ and set:
:$\ds \map I \alpha = \int_0^\infty \frac {e^{-\alpha x} } {x \sec p x} \rd x$
for all $\alpha \ge 0$.
Then:
:$\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \sec p x} \rd x = \map I a - \map I b$
We have:
{{begin-eqn}}
{{eqn | l = \map {I'} \alpha
| r = \frac \d {\d \alpha} \int_0^\infty \frac ... | :$\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \sec p x} \rd x = \frac 1 2 \map \ln {\frac {b^2 + p^2} {a^2 + p^2} }$ | Fix $p$ and set:
:$\ds \map I \alpha = \int_0^\infty \frac {e^{-\alpha x} } {x \sec p x} \rd x$
for all $\alpha \ge 0$.
Then:
:$\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \sec p x} \rd x = \map I a - \map I b$
We have:
{{begin-eqn}}
{{eqn | l = \map {I'} \alpha
| r = \frac \d {\d \alpha} \int_0^\infty ... | Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x by Secant of p x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_minus_Exponential_of_-b_x_over_x_by_Secant_of_p_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_minus_Exponential_of_-b_x_over_x_by_Secant_of_p_x | [
"Definite Integrals involving Exponential Function",
"Definite Integrals involving Cosine Function"
] | [] | [
"Definite Integral of Partial Derivative",
"Derivative of Exponential Function/Corollary 1",
"Definite Integral to Infinity of Exponential of -a x by Cosine of b x",
"Primitive of x over x squared plus a squared",
"Difference of Logarithms"
] |
proofwiki-17493 | Definite Integral to Infinity of Cosine p x minus Cosine q x over x Squared | :$\ds \int_0^\infty \frac {\cos p x - \cos q x} {x^2} \rd x = \frac {\pi \paren {\size q - \size p} } 2$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\cos p x - \cos q x} {x^2} \rd x
| r = \int_0^\infty \frac {1 - \cos q x - \paren {1 - \cos p x} } {x^2} \rd x
}}
{{eqn | r = \int_0^\infty \frac {1 - \cos q x} {x^2} \rd x - \int_0^\infty \frac {1 - \cos p x} {x^2} \rd x
}}
{{eqn | r = \frac \pi 2 \size q - \frac \pi 2 \... | :$\ds \int_0^\infty \frac {\cos p x - \cos q x} {x^2} \rd x = \frac {\pi \paren {\size q - \size p} } 2$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\cos p x - \cos q x} {x^2} \rd x
| r = \int_0^\infty \frac {1 - \cos q x - \paren {1 - \cos p x} } {x^2} \rd x
}}
{{eqn | r = \int_0^\infty \frac {1 - \cos q x} {x^2} \rd x - \int_0^\infty \frac {1 - \cos p x} {x^2} \rd x
}}
{{eqn | r = \frac \pi 2 \size q - \frac \pi 2 \... | Definite Integral to Infinity of Cosine p x minus Cosine q x over x Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Cosine_p_x_minus_Cosine_q_x_over_x_Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Cosine_p_x_minus_Cosine_q_x_over_x_Squared | [
"Definite Integrals involving Cosine Function"
] | [] | [
"Integral to Infinity of One minus Cosine p x over x Squared"
] |
proofwiki-17494 | Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x by Cosecant of p x | :$\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \csc p x} \rd x = \arctan \frac b p - \arctan \frac a p$ | Fix $p$ and set:
:$\ds \map I \alpha = \int_0^\infty \frac {e^{-\alpha x} } {x \csc p x} \rd x$
for all $\alpha \ge 0$.
Then:
:$\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \csc p x} \rd x = \map I a - \map I b$
We have:
{{begin-eqn}}
{{eqn | l = \map {I'} \alpha
| r = \frac \d {\d \alpha} \int_0^\infty \frac ... | :$\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \csc p x} \rd x = \arctan \frac b p - \arctan \frac a p$ | Fix $p$ and set:
:$\ds \map I \alpha = \int_0^\infty \frac {e^{-\alpha x} } {x \csc p x} \rd x$
for all $\alpha \ge 0$.
Then:
:$\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } {x \csc p x} \rd x = \map I a - \map I b$
We have:
{{begin-eqn}}
{{eqn | l = \map {I'} \alpha
| r = \frac \d {\d \alpha} \int_0^\infty ... | Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x by Cosecant of p x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_minus_Exponential_of_-b_x_over_x_by_Cosecant_of_p_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_minus_Exponential_of_-b_x_over_x_by_Cosecant_of_p_x | [
"Definite Integrals involving Exponential Function",
"Definite Integrals involving Sine Function"
] | [] | [
"Definite Integral of Partial Derivative",
"Derivative of Exponential Function/Corollary 1",
"Definite Integral to Infinity of Exponential of -a x by Sine of b x",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-17495 | Definite Integral to Infinity of Exponential of -a x by One minus Cosine x over x Squared | :$\ds \int_0^\infty \frac {e^{-a x} \paren {1 - \cos x} } {x^2} \rd x = \arccot a - \frac a 2 \map \ln {a^2 + 1} + a \ln a$ | Set:
:$\ds \map I a = \int_0^\infty \frac {e^{-a x} \paren {1 - \cos x} } {x^2} \rd x$
for $a > 0$.
We have:
{{begin-eqn}}
{{eqn | l = \map {I''} a
| r = \frac {\d^2} {\d a^2} \int_0^\infty \frac {e^{-a x} \paren {1 - \cos x} } {x^2} \rd x
}}
{{eqn | r = \frac \d {\d a} \int_0^\infty \frac \partial {\partial a}... | :$\ds \int_0^\infty \frac {e^{-a x} \paren {1 - \cos x} } {x^2} \rd x = \arccot a - \frac a 2 \map \ln {a^2 + 1} + a \ln a$ | Set:
:$\ds \map I a = \int_0^\infty \frac {e^{-a x} \paren {1 - \cos x} } {x^2} \rd x$
for $a > 0$.
We have:
{{begin-eqn}}
{{eqn | l = \map {I''} a
| r = \frac {\d^2} {\d a^2} \int_0^\infty \frac {e^{-a x} \paren {1 - \cos x} } {x^2} \rd x
}}
{{eqn | r = \frac \d {\d a} \int_0^\infty \frac \partial {\partia... | Definite Integral to Infinity of Exponential of -a x by One minus Cosine x over x Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_by_One_minus_Cosine_x_over_x_Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_by_One_minus_Cosine_x_over_x_Squared | [
"Definite Integrals involving Exponential Function",
"Definite Integrals involving Cosine Function"
] | [] | [
"Definite Integral of Partial Derivative",
"Derivative of Exponential Function/Corollary 1",
"Definite Integral of Partial Derivative",
"Derivative of Exponential Function/Corollary 1",
"Primitive of Exponential of a x",
"Definite Integral to Infinity of Exponential of -a x by Cosine of b x",
"Exponenti... |
proofwiki-17496 | Definite Integral of Periodic Function | Let $f$ be a Darboux integrable periodic function with period $L$.
Let $\alpha \in \R$ and $n \in \Z$.
Then:
:$\ds \int_\alpha^{\alpha + n L} \map f x \d x = n \int_0^L \map f x \d x$ | For $n \ge 0$:
{{begin-eqn}}
{{eqn | l = \int_\alpha^{\alpha + n L} \map f x \d x
| r = \int_\alpha^0 \map f x \d x + \sum_{k \mathop = 0}^{n - 1} \int_{k L}^{\paren {k + 1} L} \map f x \d x + \int_{n L}^{\alpha + n L} \map f x \d x
| c = Sum of Integrals on Adjacent Intervals for Integrable Functions/Corol... | Let $f$ be a [[Definition:Darboux Integrable Function|Darboux integrable]] [[Definition:Periodic Real Function|periodic function]] with [[Definition:Period of Periodic Real Function|period]] $L$.
Let $\alpha \in \R$ and $n \in \Z$.
Then:
:$\ds \int_\alpha^{\alpha + n L} \map f x \d x = n \int_0^L \map f x \d x$ | For $n \ge 0$:
{{begin-eqn}}
{{eqn | l = \int_\alpha^{\alpha + n L} \map f x \d x
| r = \int_\alpha^0 \map f x \d x + \sum_{k \mathop = 0}^{n - 1} \int_{k L}^{\paren {k + 1} L} \map f x \d x + \int_{n L}^{\alpha + n L} \map f x \d x
| c = [[Sum of Integrals on Adjacent Intervals for Integrable Functions/Co... | Definite Integral of Periodic Function | https://proofwiki.org/wiki/Definite_Integral_of_Periodic_Function | https://proofwiki.org/wiki/Definite_Integral_of_Periodic_Function | [
"Definite Integrals",
"Periodic Functions"
] | [
"Definition:Darboux Integrable Function",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period"
] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions/Corollary",
"General Periodicity Property",
"Integration by Substitution",
"Reversal of Limits of Definite Integral",
"Reversal of Limits of Definite Integral",
"Category:Definite Integrals",
"Category:Periodic Functions"
] |
proofwiki-17497 | Independent Subset is Contained in Maximal Independent Subset | Let $M = \struct{S, \mathscr I}$ be a matroid.
Let $A \subseteq S$.
Let $X \in \mathscr I$ such that $X \subseteq A$.
Then:
:$\exists Y \in \mathscr I : X \subseteq Y \subseteq A : \size Y = \map \rho A$
where $\rho$ is the rank function on $M$. | By definition of the rank function on $M$:
:$\size X \le \map \rho A$ | Let $M = \struct{S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $A \subseteq S$.
Let $X \in \mathscr I$ such that $X \subseteq A$.
Then:
:$\exists Y \in \mathscr I : X \subseteq Y \subseteq A : \size Y = \map \rho A$
where $\rho$ is the [[Definition:Rank Function (Matroid)|rank function]] on $M$. | By definition of the [[Definition:Rank Function (Matroid)|rank function]] on $M$:
:$\size X \le \map \rho A$ | Independent Subset is Contained in Maximal Independent Subset | https://proofwiki.org/wiki/Independent_Subset_is_Contained_in_Maximal_Independent_Subset | https://proofwiki.org/wiki/Independent_Subset_is_Contained_in_Maximal_Independent_Subset | [
"Matroid Independent Subsets"
] | [
"Definition:Matroid",
"Definition:Rank Function (Matroid)"
] | [
"Definition:Rank Function (Matroid)",
"Definition:Rank Function (Matroid)"
] |
proofwiki-17498 | Automorphic Numbers in Base 10 | If leading zeroes are allowed, there are exactly $4$ $n$-digit automorphic numbers in base $10$:
:$00 \dots 00$
:$00 \dots 01$
:$5^{2^{n - 1} } \pmod {10^n}$
:$6^{5^{n - 1} } \pmod {10^n}$ | The proof proceeds by induction on $n$.
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:There are exactly $4$ $n$-digit automorphic numbers of the forms above. | If leading [[Definition:Zero Digit|zeroes]] are allowed, there are exactly $4$ $n$-[[Definition:Digit|digit]] [[Definition:Automorphic Number|automorphic numbers]] in [[Definition:Number Base|base $10$]]:
:$00 \dots 00$
:$00 \dots 01$
:$5^{2^{n - 1} } \pmod {10^n}$
:$6^{5^{n - 1} } \pmod {10^n}$ | The proof proceeds by [[Definition:Mathematical Induction|induction]] on $n$.
For all $n \in \Z_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:There are exactly $4$ $n$-[[Definition:Digit|digit]] [[Definition:Automorphic Number|automorphic numbers]] of the forms above. | Automorphic Numbers in Base 10 | https://proofwiki.org/wiki/Automorphic_Numbers_in_Base_10 | https://proofwiki.org/wiki/Automorphic_Numbers_in_Base_10 | [
"Automorphic Numbers"
] | [
"Definition:Zero Digit",
"Definition:Digit",
"Definition:Automorphic Number",
"Definition:Number Base"
] | [
"Definition:Mathematical Induction",
"Definition:Proposition",
"Definition:Digit",
"Definition:Automorphic Number",
"Definition:Digit",
"Definition:Automorphic Number",
"Definition:Digit",
"Definition:Automorphic Number",
"Definition:Digit",
"Definition:Automorphic Number",
"Definition:Digit",
... |
proofwiki-17499 | Seventeen Horses/General Problem 1 | A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $3$ heirs.
They are to be distributed in the ratio $\dfrac 1 a : \dfrac 1 b : \dfrac 1 c$.
Let $\dfrac 1 a + \dfrac 1 b + \dfrac 1 c < 1$.
Then there are $7$ possible values of $\tuple {n, a, b, c}$ such that the required shares are:
... | It is taken as a condition that $a \ne b \ne c \ne a$.
We have that:
:$\dfrac 1 a + \dfrac 1 b + \dfrac 1 c + \dfrac 1 n = 1$
and so we need to investigate the solutions to the above equations.
From Sum of 4 Unit Fractions that equals 1, we have that the only possible solutions are:
{{begin-eqn}}
{{eqn | l = \dfrac 1 2... | A man dies, leaving $n$ indivisible and indistinguishable objects to be divided among $3$ heirs.
They are to be distributed in the [[Definition:Ratio|ratio]] $\dfrac 1 a : \dfrac 1 b : \dfrac 1 c$.
Let $\dfrac 1 a + \dfrac 1 b + \dfrac 1 c < 1$.
Then there are $7$ possible values of $\tuple {n, a, b, c}$ such that t... | It is taken as a condition that $a \ne b \ne c \ne a$.
We have that:
:$\dfrac 1 a + \dfrac 1 b + \dfrac 1 c + \dfrac 1 n = 1$
and so we need to investigate the solutions to the above equations.
From [[Sum of 4 Unit Fractions that equals 1]], we have that the only possible solutions are:
{{begin-eqn}}
{{eqn | l = \... | Seventeen Horses/General Problem 1 | https://proofwiki.org/wiki/Seventeen_Horses/General_Problem_1 | https://proofwiki.org/wiki/Seventeen_Horses/General_Problem_1 | [
"Seventeen Horses",
"Unit Fractions"
] | [
"Definition:Ratio"
] | [
"Sum of 4 Unit Fractions that equals 1"
] |
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