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linear-algebra | Connection between eigenvalues of a real matrix A and its norm. | https://math.stackexchange.com/questions/3721304/connection-between-eigenvalues-of-a-real-matrix-a-and-its-norm | <p>Is there a connection among norm of a matrix, its eigenvalues and the image of <span class="math-container">$A$</span>? Specifically if all eigenvalues of a matrix <span class="math-container">$A$</span> (<span class="math-container">$n$</span> by <span class="math-container">$n$</span>) have absolute value less tha... | <p>Let
<span class="math-container">$$A = \begin{pmatrix}1/2 & 1 \\ 0 & 1/2\end{pmatrix}$$</span>
Then <span class="math-container">$1/2$</span> is the only eigenvalue of <span class="math-container">$A$</span>. If <span class="math-container">$v = \begin{pmatrix}1 \\ 1\end{pmatrix}$</span>, then <span class="m... | 0 |
linear-algebra | Show that every function $f\ \in \mathcal{F}(F,F)$ is uniquely a polynomial of degree $\leq q−1$ - (Approach) | https://math.stackexchange.com/questions/3722183/show-that-every-function-f-in-mathcalff-f-is-uniquely-a-polynomial-of-d | <p>Let <span class="math-container">$F$</span> be a finite field with <span class="math-container">$q$</span> elements. Show that every function <span class="math-container">$f \in \mathcal{F}(F,F)$</span> is uniquely a polynomial of degree <span class="math-container">$\leq q−1$</span> with coefficients in <span class... | <p>One way of approaching this question is to analyse the natural map <span class="math-container">$\phi:F[x]\rightarrow \mathcal{F}(F,F)$</span> which interprets a polynomial as a function from <span class="math-container">$F$</span> to <span class="math-container">$F$</span>.</p>
<p>We want to say that this map, when... | 1 |
linear-algebra | Something is counted wrongly in a determinant of block matrices | https://math.stackexchange.com/questions/3722339/something-is-counted-wrongly-in-a-determinant-of-block-matrices | <p>I am counting something wrongly as I am looking at the determinant of a block matrix.</p>
<p>Let us consider this example:
<span class="math-container">$$
M = \begin{pmatrix} a & 1^T \\ 1 & I_{n-1}\end{pmatrix}
$$</span>
where <span class="math-container">$a\neq 0$</span> is scalar, <span class="math-conta... | 2 | |
linear-algebra | Dimension and sequences | https://math.stackexchange.com/questions/16490/dimension-and-sequences | <p>Consider the linear space $c^{(3)}$ of all sequences $x = (x_n)_{n=1}^{\infty}$ such that $\{x_{3k+q} \}_{k=0}^{\infty}$ converges for $q = 0,1,2$. Find the dimension and a basis for $c^{(3)}/c_0$. Note that $c_0$ is the linear space of sequences that converge to $0$.</p>
<p>I think the dimension is $1$ using the r... | <p>Notice that $c^{(3)}\cong c\oplus c \oplus c$ where $c$ is the space of convergent sequences, while $c_0 \cong c_0 \oplus c_0 \oplus c_0$ (both isomorphisms are separating the sequence to its three sub sequences modulo 3).</p>
<p>Also, in the represetation above of $c^{(3)}$ and $c_0$ we get that each copy of $c_0$... | 3 |
linear-algebra | Find a T from $R^3$ to $R^4$ given an equation for a subspace in $R^4$ | https://math.stackexchange.com/questions/19335/find-a-t-from-r3-to-r4-given-an-equation-for-a-subspace-in-r4 | <p>A subspace V in $R^4$ is defined by the equation $x_{1}$-$x_{2}$+$2x_{3}$+$4x_{4}$=0. I need to find T such that Ker(T)=zero vector, and Im(T)=V. How do I approach this problem? As I understand, the equation given to me is a set of points that are solutions to Im(T)=V, so in a sense they vectors of that plane are el... | <p>I am going to treat "matrix" and "linear transformation" as synonyms in this answer, because we are working in $\mathbb{R}^n$ and have the standard basis at our disposal. </p>
<p>Rephrasing your problem: you are given a subspace $V$ of $\mathbb{R}^4$, defined by a homogeneous linear equation. The problem is to fi... | 4 |
linear-algebra | Regarding orthonormal basis | https://math.stackexchange.com/questions/26779/regarding-orthonormal-basis | <p>I'm confronted with this question:</p>
<blockquote>
<p>Let <span class="math-container">$V$</span> be an inner product space and <span class="math-container">$B=\{u_{1}, ..., u_{n}\}$</span> a basis of <span class="math-container">$V$</span>.</p>
<p>Suppose there exists <span class="math-container">$\lambda_{1},...,... | <p>It is false. A simple example would be to take $\lambda_i = 0$, $\forall i$ or for $i$'s which are not orthogonal to the rest.</p>
<p>If you assume $\lambda_i \neq 0$, $\forall i$, then here is a counter example in $2D$.</p>
<p>Let $u_1 = 2e_1 + e_2$ and $u_2 = e_1 + 2e_2$, where $e_1, e_2$ form the conventional o... | 5 |
linear-algebra | Prove a 3x3 system of linear equations with arithmetic progression coefficients has infinitely many solutions | https://math.stackexchange.com/questions/27259/prove-a-3x3-system-of-linear-equations-with-arithmetic-progression-coefficients | <p>How can I prove that a 3x3 system of linear equations of the form:</p>
<p>$\begin{pmatrix}
a&a+b&a+2b\\
c&c+d&c+2d\\
e&e+f&e+2f
\end{pmatrix}
\begin{pmatrix}
x\\ y\\ z
\end{pmatrix}
=\begin{pmatrix}
a+3b\\
c+3d\\
e+3f
\end{pmatrix}$</p>
<p>for $a,b,c,d,e,f \in \mathbb Z$ will al... | <p>First, consider the homogeneous system
$$\left(\begin{array}{ccc}
a & a+b & a+2b\\\
c & c+d & c+2d\\\
e & e+f & e+2f
\end{array}\right)\left(\begin{array}{c}x\\y\\z\end{array}\right) = \left(\begin{array}{c}0\\0\\0\end{array}\right).$$
If $(a,c,e)$ and $(b,d,f)$ are not scalar multiples o... | 6 |
linear-algebra | Trying to find $f(x)\in F[x]$ such that $f(A)=A^{-1}$ | https://math.stackexchange.com/questions/29580/trying-to-find-fx-in-fx-such-that-fa-a-1 | <p>Given an invertible $3\times 3$ matrix:</p>
<p>$A = \begin{pmatrix}
1 & 2 & 2 \\
1 & 2 & -1 \\
-1 & 1 & 4
\end{pmatrix}$</p>
<p>I am trying to find $f(x)$ from $F[x]$ such that $A^{-1}=f(A)$. To do so, I want to use the result of <a href="https://math.stackexchange.com/questions/29158">... | <p>You can use the following two facts:</p>
<ul>
<li>Every square matrix is a zero of its characteristic polynomial.</li>
<li>The constant term of the characteristic polynomial of a matrix is its determinant.</li>
</ul>
<p>Combining these two things you can write the characteristic polynomial $c_A(x) = x \cdot p(x) +... | 7 |
linear-algebra | Help understanding this example of a linear operator which rotates each vector $v$ about the z-axis by an angle $\theta$ | https://math.stackexchange.com/questions/49267/help-understanding-this-example-of-a-linear-operator-which-rotates-each-vector | <blockquote>
<p>Let <span class="math-container">$T: \mathbb{R}^{3} \to \mathbb{R}^{3}$</span> be the following linear operator, which rotates each vector <span class="math-container">$v$</span> about the <span class="math-container">$z$</span>-axis by an angle <span class="math-container">$\theta$</span>: <span class=... | <p>First, the number $\theta$ is a parameter which you should think of as some number fixed for all time (or, it's "on the side" as you put it). The function from $\mathbb{R}^4\rightarrow\mathbb{R}^3$ you described is <em>not</em> linear in $\theta$.</p>
<p>Second, the formulas $x\cos\theta - y\sin\theta$ and $x\sin\... | 8 |
linear-algebra | On a matrix factorization and the Gram-Schmidt process | https://math.stackexchange.com/questions/67803/on-a-matrix-factorization-and-the-gram-schmidt-process | <p>Given a real square matrix <span class="math-container">$A$</span>, we can factor it as <span class="math-container">$$A = QR$$</span> where <span class="math-container">$Q$</span> is orthogonal and <span class="math-container">$R$</span> is upper triangular. The entries of <span class="math-container">$R$</span> ha... | 9 | |
linear-algebra | Finding the matrix of this linear transformation | https://math.stackexchange.com/questions/91324/finding-the-matrix-of-this-linear-transformation | <blockquote>
<p>We're given <span class="math-container">$V$</span>, which is an <span class="math-container">$n$</span> dimensional vector space. <span class="math-container">$T : V \to V$</span> is a linear transformation. There is a vector <span class="math-container">$v \in V$</span> such that <span class="math-con... | <p>Yes, $T^2(v)$ means $T(T(v))$ in general $T^k(v)$ is composition of $T$ with itself $k$ times. And you are correct about how we should go about this problem, compute how the transformation acts on the basis. I will describe the process for how to do the first part below, and to do the second part you will do the exa... | 10 |
linear-algebra | Find values for $a$, $b$, $c$ that make this linear system solvable? | https://math.stackexchange.com/questions/213580/find-values-for-a-b-c-that-make-this-linear-system-solvable | <p>I came along with the following exercise that I developed poorly. May anybody give me some light? See:</p>
<blockquote>
<p>How to find a solution involving <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$c$</span> to make the following system consiste... | <p>The very first step in row reduction gives the new equation
$$ -5y -2z -4t = c-3a$$
Since the left-hand side of this is minus the left-hand side of the equation for $b$, the equations imply $b=3a-c$.</p>
<p>On the other hand, that same first step of row reduction shows that the rank of the equation system is at lea... | 11 |
linear-algebra | Linear Algebra (linear transformations) | https://math.stackexchange.com/questions/255298/linear-algebra-linear-transformations | <blockquote>
<p>13. Suppose <span class="math-container">$V$</span> and <span class="math-container">$W$</span> are finite-dimensional vector spaces and <span class="math-container">$T:V \to W$</span> is an isomorphism. Then there exist bases <span class="math-container">$\mathcal{B}$</span> and <span class="math-conta... | <p>For 14 recall that $\mathbb{R}$ is a vector space over itself with dimension $1$ . what is $dim(Im(T))$ ?</p>
<p>For 15 recall there is a matrix $A$ s.t $Tv=Av$ for all $v$.</p>
| 12 |
linear-algebra | Linear Algebra Proof | https://math.stackexchange.com/questions/255898/linear-algebra-proof | <blockquote>
<p>If A is a <span class="math-container">$m\times n$</span> matrix and <span class="math-container">$M = (A \mid b)$</span> the augmented matrix
for the linear system <span class="math-container">$Ax = b$</span>.</p>
<p>Show that either<br> <br><span class="math-container">$(i) \operatorname{rank}A = \ope... | <p>Suppose the columns of $A$ have exactly $r$ linearly independent vectors. If $b$ lies in their span, then $\operatorname{rank} A=r=\operatorname{rank} M$. If not, then the columns of $A$ together with $b$ have exactly $(r+1)$ linearly independent vectors, so that $\operatorname{rank} A+1=r+1=\operatorname {rank} M$.... | 13 |
linear-algebra | Question on a proof about the Rank of a Matrix | https://math.stackexchange.com/questions/301629/question-on-a-proof-about-the-rank-of-a-matrix | <p>The question is:</p>
<blockquote>
<p>Give a formal proof for the following statement:
Given a matrix A and a scalar c, show that rank(cA) = rank(A)</p>
</blockquote>
<p>Here are the steps that I took to go about the proof:</p>
<blockquote>
<p>(1) Prove this claim: Let v1, v2, ..., vN be vectors</p>
<p>then {v1, v2, ... | <p>The rank is the dimension of the range.</p>
<p>Now try to prove that for all $c\neq 0$, the range of $A$ is equal to the range of $cA$.</p>
<p>Hints:
$$
cA(x)=A(cx)\quad\mbox{and}\quad A(x)=cA\left( \frac{1}{c}x\right).
$$</p>
<p>This is easier like this.</p>
| 14 |
linear-algebra | Dual Space Questions | https://math.stackexchange.com/questions/354235/dual-space-questions | <blockquote>
<p>Let <span class="math-container">$V$</span> be a finite dimensional vector space over a field <span class="math-container">$F$</span>.</p>
<p>Let <span class="math-container">$v\in V$</span> with <span class="math-container">$v$</span> not equal to <span class="math-container">$0$</span>. Show that ther... | <p><strong>Hint:</strong> Extend $v$ to a basis of $V$, say $\{v=v_1, v_2,v_3,\dots v_n\}$. A function $T:V\rightarrow F$is linear iff $T\bigg(\sum_{i=1}^n\lambda_iv_i\bigg) =\sum_{i=1}^n\lambda_iT(v_i)$ for any $\lambda_i \in F$. Note that $T$ is uniquely determined by its action on a basis, so defining the values of ... | 15 |
linear-algebra | If the union of $A$ and $B$ is linearly independent then the intersection of the spans $= \{0\}$ | https://math.stackexchange.com/questions/392636/if-the-union-of-a-and-b-is-linearly-independent-then-the-intersection-of-the | <blockquote>
<p><span class="math-container">$\newcommand{\sp}{\operatorname{sp}}$</span> Let <span class="math-container">$V$</span> be a vector space over <span class="math-container">$F$</span> field, and let <span class="math-container">$A,B$</span> be two different, disjoint, non-empty sets of vectors from <span c... | <p>Hint:</p>
<p>$$x\in Sp(A)\cap Sp(B)\implies \exists\,a_1,...,a_k\in A\;,\;b_1,...,b_m\in B\,,\,c_1,...,c_{k+m}\in \Bbb F\;\;s.t.$$</p>
<p>$$x=\sum_{i=1}^kc_ia_i=\sum_{i=1}^mc_{k+1}b_i\implies c_1a_1+\ldots c_ka_k-c_{k+1}b_1-\ldots c_mb_m=0\implies$$</p>
<p>$$\implies c_r=0\;\;\forall r=1,\ldots,m\;,\;\text{since}... | 16 |
linear-algebra | Projection and inner product space | https://math.stackexchange.com/questions/397065/projection-and-inner-product-space | <p>Definition: Let <span class="math-container">$V$</span> be vector space, and <span class="math-container">$U$</span>, <span class="math-container">$W$</span> be two subspaces such that <span class="math-container">$V=U\oplus W$</span>.</p>
<p>We know that there exists for each <span class="math-container">$v \in V$<... | <p>Hint: you can prove $P_U=P_{U,U^\perp}$ by checking that $P_U(v)=P_{U,U^\perp}(v)$ for every $v\in V$.</p>
<hr>
<p>Added: You've got the start of the right strategy, but let me modify it it a bit. Let's start with your $v=u+w$ with $u\in U$ and $w\in U^\perp$ (I think you might be forgetting about this last fact.)... | 17 |
linear-algebra | On $C^0 [0, 1]$, define $f \cdot g = \int_0^1 f(x) g(x) dx$. For $f(x) = x$. | https://math.stackexchange.com/questions/561785/on-c0-0-1-define-f-cdot-g-int-01-fx-gx-dx-for-fx-x | <blockquote>
<p>a. find <span class="math-container">$||f||$</span></p>
<p>b. find all linear polynomials that are orthogonal to <span class="math-container">$x$</span></p>
</blockquote>
<p>Okay, so I know that</p>
<p><span class="math-container">$||f|| = \sqrt(f_1^2 + f_2^2 +... + f_n^2)$</span></p>
<p>and that linear... | <p>If you have a scalar product $(f,g)$ then the norm is defined by $\|f\| = \sqrt{(f,f)}$. In your case
$$ \|f\|^2 = \int_0^1 f(x)^2 dx$$</p>
<p>To find $\|f\|$ you just need to integrate $x^2$ from $0$ to $1$.</p>
<p>If you have a linear polynomial $ax+b$ which is orthogonal to $x$ then their scalar product is zer... | 18 |
linear-algebra | Prove or disprove: If $Null(A-B)=\mathbb R^n$ then $ A=B $ | https://math.stackexchange.com/questions/617541/prove-or-disprove-if-nulla-b-mathbb-rn-then-a-b | <blockquote>
<p><span class="math-container">$A$</span> and <span class="math-container">$B$</span> are matrices of order <span class="math-container">$m\times n$</span>.</p>
<p>Prove or disprove: If <span class="math-container">$Null(A-B)=\mathbb R^n$</span> then <span class="math-container">$ A=B $</span></p>
</block... | <p>You might try by contrapositive: if $A\neq B$, then there must be some vector$~v$ such that $Av\neq Bv$. Then $(A-B)v\neq\ldots$</p>
<p>(continued) $\ldots\neq0$, since $(A-B)v=Av-Bv$ by definition. So $\def\Null{\operatorname{Null}}v\notin\Null(A-B)$ which proves $\Null(A-B)\neq\Bbb R^n$. We have shown $A\neq B\i... | 19 |
linear-algebra | Proving that a set of functions is a vector space | https://math.stackexchange.com/questions/984816/proving-that-a-set-of-functions-is-a-vector-space | <p>We've given that <span class="math-container">$V$</span> is a vector space and that <span class="math-container">$L(V)$</span> the set with functions <span class="math-container">$T:V\rightarrow \mathbb{R}$</span> s.t. <span class="math-container">$T(a_1f_1+a_2f_2)=a_1T(f_1)+a_2T(f_2)$</span>. We must show that <spa... | 20 | |
linear-algebra | Homogeneous system of equations , and sub-set K of $R^4$ | https://math.stackexchange.com/questions/852442/homogeneous-system-of-equations-and-sub-set-k-of-r4 | <p>Given K,L are sub-sets of <span class="math-container">$K^4$</span>:</p>
<p><span class="math-container">$K = \{(-5,8,14,0),(-1,4,2,4)\}, L = \{(0,1,-10,8),(0,3,-1,5)\}$</span></p>
<blockquote>
<p>Find a homogeneous system of equations that its solutions are Spanned by K.</p>
<p>Also prove that L spans the solutions... | <p>Both span(K) and span(L) are two-dimensional, so in $\mathbb{R}^4$ we expect to have two linear equations (since # of equations + # dimensions = dimension of larger vector space). Logically, there should be infinitely many possible pairs of equations that work, similar to how a line in 3D space has infinitely many p... | 21 |
linear-algebra | If $AB = 0$, prove that the columns of matrix $B$ are vectors in the kernel of $A$ | https://math.stackexchange.com/questions/1095451/if-ab-0-prove-that-the-columns-of-matrix-b-are-vectors-in-the-kernel-of | <blockquote>
<p>Let <span class="math-container">$A,B$</span> be <span class="math-container">$n\times n$</span> matrices.</p>
<p>If <span class="math-container">$AB=0$</span>, prove that the columns of matrix <span class="math-container">$B$</span> are vectors in the kernel of <span class="math-container">$Ax=0$</span... | <p><strong>Hint:</strong></p>
<p>$$ AB = \begin{pmatrix} Ab^1 && ... && Ab^n \end{pmatrix} $$</p>
<p>Where $b^i$ is the i-th column vector and the right side is the matrix you get by the multiplication.</p>
<p><strong>Details:</strong>
By comparing the two matrices you can now conclude that for all t... | 22 |
linear-algebra | How do I find which set of functions is linearly independent? | https://math.stackexchange.com/questions/1097550/how-do-i-find-which-set-of-functions-is-linearly-independent | <blockquote>
<p>Choose the correct set of functions, which are not linearly independent.</p>
<ol>
<li><span class="math-container">$x^2-1$</span>, <span class="math-container">$2x^2-x+1$</span>, <span class="math-container">$3x^2-x$</span></li>
<li><span class="math-container">$1$</span>, <span class="math-container">$... | <p>Since
$3x^2-x-(x^2-1)=2x^2-x+1$</p>
<p>so the first set of vectors is linearly dependent.</p>
<p>For the rest find the <a href="http://mathworld.wolfram.com/Wronskian.html" rel="nofollow">Wronskian</a> </p>
<p>For example </p>
<p>$W(x^2,x^3,x^4)=6x^6$, thus the set $\{x^2,x^3,x^4\}$ is linearly independent if an... | 23 |
linear-algebra | Projections onto a subspace (orthogonal vs. non-orthogonal matrix vs. basis matrix) | https://math.stackexchange.com/questions/1303925/projections-onto-a-subspace-orthogonal-vs-non-orthogonal-matrix-vs-basis-matr | <p>Suppose</p>
<blockquote>
<p><span class="math-container">$A$</span> is our matrix</p>
<p><span class="math-container">$B$</span> is our basis for the matrix <span class="math-container">$A$</span></p>
<p><span class="math-container">$Q$</span> is orthogonal basis for matrix <span class="math-container">$A$</span></p... | 24 | |
linear-algebra | understanding a linear transformation | https://math.stackexchange.com/questions/779180/understanding-a-linear-transformation | <p>Hello I'm trying to solve this question from Ron Larson's linear algebra textbook. But I'm just stuck on how to approach this question. Could someone please at least give me a hint on how to approach this sort of question?</p>
<blockquote>
<p>Suppose <span class="math-container">$T:\unicode{x211D}^2\rightarrow\unico... | <p>Note that $(x,y)=x(1,0)+y(0,1)$. Therefore, because $T$ is linear, $$T(x,y)=xT(1,0)+yT(0,1)=x(0,1)+y(1,0)=(y,x)$$
So $T$ maps the point $(x,y)$ to the point $(y,x)$. What does this do geometrically?</p>
| 25 |
linear-algebra | Dimension Theorem Corollary | https://math.stackexchange.com/questions/1198165/dimension-theorem-corollary | <p>Let $V$ and $W$ be vector spaces with $\dim V = \dim W$. If $T : V → W$ is
linear then $T$ is one-to-one if and only if $T$ is onto.
But this is true only when the dimensions of $V$ and $W$ are finite.
For instance I came across the example $T : P(R)\to P(R)$ such that $T(f(x))=f'(x)$. Here T is onto but not one-on... | <p>Firstly, note that without the <a href="http://en.wikipedia.org/wiki/Axiom_of_choice" rel="nofollow noreferrer">axiom of choice</a>, we can't speak about the dimension of a vector space in general, because <a href="https://math.stackexchange.com/a/207992/26369">there would be vector spaces without a basis</a>! For t... | 26 |
linear-algebra | Does $ABC=D\implies \det(ABC)=\det(D )$? | https://math.stackexchange.com/questions/1332815/does-abc-d-implies-detabc-detd | <p><span class="math-container">$${\color{brown}{\text{Question I am trying to solve:}}}$$</span></p>
<p>Let <span class="math-container">$A,B$</span> and <span class="math-container">$X$</span> be 7 x 7 matrices such that <span class="math-container">$\det A=1$</span>, <span class="math-container">$\det B=3$</span> an... | <p>I might be missing something here, but as far as I know:</p>
<p>$a = a' \Rightarrow f(a) = f(a')$</p>
<p>for all sets $A,B$; $a,a'\in A$ and functions $f : A\to B$.</p>
| 27 |
linear-algebra | About the matrix of two linear transformations | https://math.stackexchange.com/questions/704052/about-the-matrix-of-two-linear-transformations | <p>I have an exercise to answer, and I don't know if I've done it the right way. This is only a little part of the exercise, but I have to know if what I've done so far is correct. Here we go:</p>
<p>Let $V$ be a $K$-vector space and $\dim(V)=4$. Let $B_{1}=(u_1,u_2,u_3,u_4)$ be a basis of $V$.
Let $W$ be a $K$-vector... | <p>You can find the matrix of the maps with respect to the bases by both ways: if you know how coordinates of the result can be obtained from the coordinates of an input, and if you know how elements of the basis in the domain are mapped to vectors in the codomain (that is what you have done).</p>
<p>This is a matter ... | 28 |
linear-algebra | Linear Independence and Subset Relations | https://math.stackexchange.com/questions/1418113/linear-independence-and-subset-relations | <p>I've been reading the <a href="https://en.wikibooks.org/wiki/Linear_Algebra/Definition_and_Examples_of_Linear_Independence" rel="nofollow noreferrer">wikibook</a> on Linear Algebra and in the section 'Linear Independence and Subset Relations' it defines the following lemma:</p>
<blockquote>
<p>Lemma 1.14:
Any subset... | <p>There are several mistakes in your proof. First of all, it is not clear what you want to say in your first line. You can assume
$$S= \{v_1, \dots, v_n\}\space$$
and assume it is linearly independent. But that does not tell you the equation you wrote. </p>
<p>I suppose you want to show it by contradiction. Since th... | 29 |
linear-algebra | If a lower triangular matrix is nonsingular, then its inverse is also lower triangular | https://math.stackexchange.com/questions/1425984/if-a-lower-triangular-matrix-is-nonsingular-then-its-inverse-is-also-lower-tria | <p>I already have the result that says that if <span class="math-container">$U$</span> is upper triangular and non singular then <span class="math-container">$U^{-1}$</span> is also upper triangular. I want to use this result to prove the result for lower triangular matrix <span class="math-container">$n \times n$</spa... | 30 | |
linear-algebra | Prove that there exists a matrix $B$ s.t ker$B=$Im$A$, Im$B=$ker$A$ | https://math.stackexchange.com/questions/1566424/prove-that-there-exists-a-matrix-b-s-t-kerb-ima-imb-kera | <blockquote>
<p>Let <span class="math-container">$A$</span> be a square matrix.</p>
<p>a) Show that there always exists a square matrix B such that Ker <span class="math-container">$B =$</span> Im <span class="math-container">$A$</span> and Ker<span class="math-container">$A =$</span> Im<span class="math-container">$B$... | <p>It is probably better to think in terms of linear maps instead of matrices. So you have a linear map $a$ on a vector space $V$. Let $v_{1}, \dots , v_{n}$ be a basis of $V$ such that $v_{1}, \dots, v_{k}$ are a basis of $\ker(a)$, so that $f(v_{k+1}), \dots , f(v_{n})$ is a basis of the image of $a$. Extend the $f(v... | 31 |
linear-algebra | Find a basis for $U\cap V$ | https://math.stackexchange.com/questions/1398300/find-a-basis-for-u-cap-v | <blockquote>
<p>Let <span class="math-container">$$a = (0,2,3,-1)^T \quad b=(0,2,7,-2)^T \quad c = (0,-2,1,0)^T \quad u = (1,2,0,1)^T\quad v = (2,2,1,2)^T$$</span>
Let <span class="math-container">$U= \langle a,b,c \rangle, V = \langle u,v\rangle$</span></p>
<p>Then a) find a basis for <span class="math-container">$U$<... | <p>Now you have two ways of expressing your vector $x$ in terms of a single arbitrary value $r$. Since $x$ was arbitrary any element of the intersection has this form. So use one of the two ways to find a fixed vector $w$ such that your arbitrary $x = rw$ for some $r$. As a check to your work so far, you should get the... | 32 |
linear-algebra | linear transformation that Im(T)=Ker(T) | https://math.stackexchange.com/questions/1429613/linear-transformation-that-imt-kert | <blockquote>
<p>Let <span class="math-container">$T:\mathbb{R}^2\rightarrow \mathbb{R}^2$</span> such that <span class="math-container">$T(x,y)=(2x-3y,\alpha x+\beta y)$</span> and <span class="math-container">$Ker(T)=Im(T)$</span></p>
<p>find <span class="math-container">$\alpha,\beta$</span></p>
</blockquote>
<p>How... | <p>You know that $T(1,0)=(2,\alpha)$ and $T(0,1)=(-3,\beta)$ belong to $\operatorname{Im} T$. Thus they also belong to $\operatorname{Ker} T$.</p>
<p>What can you deduce from the fact that $T(2,\alpha)=(0,0)$? Similarly, can you say something about $\beta$ from $T(-3,\beta)=(0,0)$?</p>
| 33 |
linear-algebra | IF $A$ is similar to $B$, then $A^{-1}$ is similar to $B^{-1}$ | https://math.stackexchange.com/questions/1491948/if-a-is-similar-to-b-then-a-1-is-similar-to-b-1 | <p>Suppose <span class="math-container">$A$</span> is similar to <span class="math-container">$B$</span> (That is: there is some nonsingular <span class="math-container">$C$</span> such that <span class="math-container">$B = C^{-1} A C $</span>). If <span class="math-container">$A$</span> is nonsingular, show that <spa... | <p>Right idea, but $(C^{-1} A C )^{-1} = C^{-1} A^{-1} C$ might need to be justified a little more. You can't just distribute the exponent if that's what you were doing. It's not hard to show that you take the inverse of a product of matrices in reverse order, and that $(C^{-1})^{-1}=C$.</p>
<p>Otherwise, this is fine... | 34 |
linear-algebra | Proving that a matrix is skew Hermitian | https://math.stackexchange.com/questions/1496710/proving-that-a-matrix-is-skew-hermitian | <p>Suppose <span class="math-container">$A \in \mathbb{C}^{n \times n} $</span> is skew hermitian: <span class="math-container">$A^* = -A$</span>. Suppose <span class="math-container">$B$</span> is unitarily similar to <span class="math-container">$A$</span>: That is there is some unitary matrix <span class="math-conta... | 35 | |
linear-algebra | Basis and dimensions | https://math.stackexchange.com/questions/1459238/basis-and-dimensions | <p>How do i find the a basis and dimension for $A[x]$?</p>
<p>Consider the subset of $R[x]$ given by $A[x]:=\{q(x)$ element of $\mathbb R_4[x]$ such that $q(2)=0=q(-3)\}$</p>
<p>I'm a bit confused because there are two conditions to be satisifed, $q(2)=0=q(-3)$</p>
| <p>I would do the following: one knows that a polynomial vanishes on a point $a$ iff it is a multiple of $(x-a)$. Because $(x-2)$ and $(x+3)$ are coprime polynomials, a polynomial vanishes on both $2$ and $-3$ iff it is a multiple of $(x-2)(x+3)$.</p>
<p>So, the subspace you're looking at is the space of polynomials o... | 36 |
linear-algebra | Explanation of notation - Linear Algebra | https://math.stackexchange.com/questions/1522119/explanation-of-notation-linear-algebra | <p>I'm reading the following piece of text:</p>
<blockquote>
<p>Let <span class="math-container">$T: V \to W$</span> and <span class="math-container">$S: U \to V$</span> be two linear transformations between vector spaces <span class="math-container">$U, V, W$</span> of finite dimension.</p>
<p>Since <span class="math-... | <p>From context and common sense I am almost sure that $R(T)$, for example, refers to the range of a linear map $T$.</p>
<p>In Friedberg's linear algebra, for instance, use $N(T)$ to denote the zero set of $T$ and $R(T)$ the range of $T$.</p>
| 37 |
linear-algebra | Trying to establish a norm inequality | https://math.stackexchange.com/questions/1469821/trying-to-establish-a-norm-inequality | <p>Let <span class="math-container">$A$</span> be <span class="math-container">$n $</span> by <span class="math-container">$n$</span> matrix and say <span class="math-container">$A = LU $</span> is the LU factorization of <span class="math-container">$A$</span>. Suppose <span class="math-container">$|l_{ij}| \leq 1 $</... | <p>Let $A$ be an $n\times n$ matrix and suppose that $A$ has an LU factorization $A = LU$. Since $L$ is invertible and $\|\cdot\|_\infty$ is submultiplicative it follows that</p>
<p>$$
\|L^{-1}A\|_\infty = \|U\|_\infty \implies \|U\|_\infty \leq \|L^{-1}\|_\infty\|A\|_\infty
$$</p>
<p>By the process of Gaussian elimi... | 38 |
linear-algebra | Consider the basis $B = \{(1, 2), (3, 4)\}$. Suppose $[x]_B =(7, 11)$ for some $x \in \mathbb R^2.$ Find $x$ with respect to $\mathcal E_2.$ | https://math.stackexchange.com/questions/1610907/consider-the-basis-b-1-2-3-4-suppose-x-b-7-11-for-some | <blockquote>
<p><span class="math-container">$\mathcal E_i$</span> denotes the standard basis.</p>
<p><span class="math-container">$[x]_B$</span> denotes the the coordinate vector with respect to the basis <span class="math-container">$B$</span>.</p>
</blockquote>
<p><span class="math-container">$a(1, 0) + b(0, 1) = (x... | <p>IMO it's best not to think of your vectors as elements of $\Bbb R^2$ -- I just don't find there to be any motivation for change of basis in $\Bbb R^n$. Instead just take your vectors as abstract objects which follow some easy rules and let the algebra take care of everything.</p>
<p>First we need to give some name... | 39 |
linear-algebra | Linear operator and its corresponding matrix. | https://math.stackexchange.com/questions/1828644/linear-operator-and-its-corresponding-matrix | <blockquote>
<p>There's linear operator <span class="math-container">$A: \mathbb{R}_2[x] \to \mathbb{R}_2[x]$</span> defined as <span class="math-container">$(A(p))(x):=p'(x+1)$</span>.</p>
<p>Find all possible values for <span class="math-container">$a, b, c \in \mathbb{R}$</span> for which matrix <span class="math-co... | <p>The matrix of your operator (which I'll rename to $T$ in order not to get confused with matrices) with respect to the basis $\mathcal{B} = (1,x,x^2)$ is given by</p>
<p>$$ [T]_{\mathcal{B}} = A = \begin{pmatrix} 0 & 1 & 2 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{pmatrix}. $$</p>
<p>Call your other ma... | 40 |
linear-algebra | How to solve linear equation using inversion method? | https://math.stackexchange.com/questions/1559659/how-to-solve-linear-equation-using-inversion-method | <p>I do not understand the inversion method to solve a pair of linear equations:</p>
<blockquote>
2x<sub>1</sub> + 4x<sub>2</sub> = 4<br>
9x<sub>1</sub> + 3x<sub>2</sub> = 6
</blockquote>
<p>How to solve this? Please clarify steps.</p>
| <p>Write in a matrix form $Ax=b$, i.e.
$$\pmatrix{2&4\\9&3}\pmatrix{x_1\\x_2}=\pmatrix{4\\6}$$
Using the following formula of inverse matrix
$$\pmatrix{a&b\\c&d}^{-1}=\frac{1}{a d-bc}\pmatrix{d&-b\\-c&a}$$
one get
$$x=\pmatrix{x_1\\x_2}=A^{-1}b=\frac{1}{2\cdot3-9\cdot4}\pmatrix{3&-4\\-9&... | 41 |
linear-algebra | Solve the system (3) | https://math.stackexchange.com/questions/1839440/solve-the-system-3 | <blockquote>
<p>Solve the system</p>
<p><span class="math-container">$x_1 + x_2 -3x_3 = -2$</span></p>
<p><span class="math-container">$4x_1 + 3x_2 + 3x_3 = 2$</span></p>
<p><span class="math-container">$\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} = \begin{bmatrix}\\\\\\\end{bmatrix} + \begin{bmatrix}\\\\\\\end{bmatrix} ... | <p>The most straight-forward way about solving this is to take the 2 equations and set them up as an <a href="https://en.wikipedia.org/wiki/Augmented_matrix" rel="nofollow">Augmented Matrix</a> and get <em>RREF</em> like so:</p>
<p>$\left[\begin{array}{ccc|c}
1 & 1 & -3 & -2\\
4 & 3 & 3 &2
\en... | 42 |
linear-algebra | Which of the following subsets of $\mathbb{R}^{3\times3}$ are subspaces of $\mathbb{R}^{3\times3}$ | https://math.stackexchange.com/questions/1980454/which-of-the-following-subsets-of-mathbbr3-times3-are-subspaces-of-mat | <blockquote>
<p>Which of the following subsets of <span class="math-container">$\mathbb{R}^{3 \times 3}$</span> are subspaces of <span class="math-container">$\mathbb{R}^{3 \times 3}$</span>?</p>
<p>A. The <span class="math-container">$3 \times 3$</span> matrices with determinant 0<br/>
B. The <span class="math-contain... | <p>(B) is false, since there are $\;3\times3\;$ integer matrices which multiplied by $\;\frac12\;$ aren't integer anymore (example?)</p>
<p>$$$$</p>
| 43 |
linear-algebra | Bases for space of polynomials | https://math.stackexchange.com/questions/1756963/bases-for-space-of-polynomials | <p>I'm facing an exercise to determine basis for some spaces of polynomials. Here they are</p>
<blockquote>
<p>Consider the space of polynomials of degree equal or less than 3</p>
<p><span class="math-container">$U =$</span>{<span class="math-container">$p(t) \in \mathbb{R_3}[t]$</span> | <span class="math-container">$... | <p>The maps $f,g,h\colon \mathbb{R}_3[x]\to\mathbb{R}$ defined by
\begin{align}
f(p)&=p(0)\\
g(p)&=p(1)\\
h(p)&=p(1)-p(0)
\end{align}
are easily seen to be linear and surjective. So their kernels (null spaces) have dimension $3$. The kernels are precisely the subspaces you have to find bases of, in the same... | 44 |
linear-algebra | If you add row $1$ of $A$ to row $2$ to get $B$, how do you find ${ B }^{ -1 }$ from ${ A}^{ -1 }$? | https://math.stackexchange.com/questions/1765374/if-you-add-row-1-of-a-to-row-2-to-get-b-how-do-you-find-b-1 | <blockquote>
<p>If you add row <span class="math-container">$1$</span> of <span class="math-container">$A$</span> to row <span class="math-container">$2$</span> to get <span class="math-container">$B$</span>, how do you find <span class="math-container">${ B }^{ -1 }$</span> from <span class="math-container">${ A}^{ -1... | <p>The inverse of $AB$ is the reverse product ${ B }^{ -1 }{ A }^{ -1 }$.</p>
<p>So by applying this to $B=\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} A \end{bmatrix}$, we get </p>
<p>$${ B }^{ -1 }={ A }^{ -1 }{ \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} }^{ -1 }$$</p>
<p>$$\Righta... | 45 |
linear-algebra | Give the matrix for the transformation T: | https://math.stackexchange.com/questions/1840777/give-the-matrix-for-the-transformation-t | <blockquote>
<p><span class="math-container">$a)$</span>Give the matrix for the Transformation <span class="math-container">$T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$</span> that first reflects points through the <span class="math-container">$x$</span> - axis and then reflects through the line <span class="math-contain... | <p>It suffices to keep track of where $(1,0)$ and $(0,1)$ are mapped to. In particular, we have
$$
(1,0) \mapsto (1,0) \mapsto (0,1)\\
(0,1) \mapsto (0,-1) \mapsto (-1,0)
$$
So, the matrix of $T$ is the matrix with these columns. That is,
$$
[T] = \pmatrix{0&-1\\1&0}
$$
It is worth noting that this is in fac... | 46 |
linear-algebra | Prove, that for every pair $(a,b)$ there is another pair $(u,v)$ so that $a\cdot v = b \cdot u$. $a,b,c,d \in \mathbb{Z}^*$ | https://math.stackexchange.com/questions/2048645/prove-that-for-every-pair-a-b-there-is-another-pair-u-v-so-that-a-cdot | <blockquote>
<p><span class="math-container">$\forall (a,b) a,b\in \mathbb{Z}^* \exists (u,v) u,v \in \mathbb{Z}^*:a\cdot v = b\cdot u$</span> <span class="math-container">$ \land gcd(u,v) =1 $</span></p>
<p>So basically I have to prove, that for every pair <span class="math-container">$(a,b)$</span> there is another ... | <p>Hint: Try $u=a/d$, $v=b/d$ with $d=...$</p>
| 47 |
linear-algebra | Prove that $Av_i \bullet Av_j = v_i \bullet v_j$, $\forall i,j$ | https://math.stackexchange.com/questions/1720707/prove-that-av-i-bullet-av-j-v-i-bullet-v-j-forall-i-j | <p>There are <span class="math-container">$2$</span> different bases in <span class="math-container">$R^2$</span>, <span class="math-container">$\{u_1,u_2\} , \{v_1,v_2\}$</span>. and <span class="math-container">$A$</span> is a matrix <span class="math-container">$nxn$</span>.</p>
<p>Is it possible to prove that</p>
<... | <p>Express $v_i=a_{i1}u_1+a_{i2}u_2$. Now by using that $\bullet$ is bilinear, i.e. $(a+b)\bullet c=a\bullet c+b\bullet c$ we get:</p>
<p>\begin{eqnarray}
Av_i\bullet Av_j&=&(a_{i1}Au_1+a_{i2}Au_2)\bullet (a_{j1}Au_1+a_{j2}Au_2)\\
&=& a_{i1}Au_1\bullet a_{j1}Au_1+a_{i2}Au_2\bullet a_{j1}Au_1+a_{i1}Au_1... | 48 |
linear-algebra | Question on how to prove that a vector space is linear | https://math.stackexchange.com/questions/1777891/question-on-how-to-prove-that-a-vector-space-is-linear | <p>This is a past exam question that wasn't explained in my lecture notes:</p>
<blockquote>
<p>For vector spaces <span class="math-container">$U$</span> and <span class="math-container">$V$</span> over the same field of scalars <span class="math-container">$\mathbb{F}$</span></p>
<p>Let <span class="math-container">$U ... | <p>Hint:</p>
<p>consider
$$
\phi(f+\alpha g)=(x+3)\frac{d}{dx}(f+\alpha g)+2
$$
where $\alpha\in \mathbb{K}$ and verify if it is an element of $P^{25}$ if $f,g \in P^{25}$ and if it is the same as:
$$
\phi(f)+\alpha \phi(g)=(x+3)\frac{df}{dx}+2+\alpha[(x+3)\frac{dg}{dx}+2 ]
$$</p>
<p>This proves linearity.</p>
| 49 |
linear-algebra | How does $A^{-1}Ax = b$ turn into $x = A^{-1}b$? | https://math.stackexchange.com/questions/1861830/how-does-a-1ax-b-turn-into-x-a-1b | <blockquote>
<p><strong>Background</strong>: Looking at properties and dealing with <a href="https://en.wikipedia.org/wiki/Matrix_(mathematics)" rel="nofollow noreferrer">Matrices</a> in linear algebra, and reading about <a href="http://mathworld.wolfram.com/MatrixInverse.html" rel="nofollow noreferrer">Matrix Inverse(... | <p>This looks like a typo to me.</p>
<p>What is true is that if $A$ is a matrix whose inverse $A^{-1}$ exists, we have</p>
<p>$$\begin{align*}Ax = b &\iff A^{-1} Ax = A^{-1} b \\
&\iff Ix = A^{-1}b\\
&\iff x = A^{-1}b\\\end{align*}$$</p>
<p>This is the matrix version of the usual rule with numbers: $ax ... | 50 |
linear-algebra | Finding formulas for the entries of a matrix | https://math.stackexchange.com/questions/1863817/finding-formulas-for-the-entries-of-a-matrix | <blockquote>
<p>Let <span class="math-container">$M = \begin{bmatrix}8&2\\-1&5\end{bmatrix}$</span> Find formulas for the entries of <span class="math-container">$M^n$</span> where <span class="math-container">$n$</span> is a positive integer</p>
<p><span class="math-container">$M^n = ?$</span> (Should be a <sp... | <p>Often when you want to take a high power of a matrix $A$, you do what's called diagonalization. That is, you find two matrices $M$ and $D$ where $D$ is diagonal and $A = M D M^{-1}$. Then, we have that $A^n = (M D M^{-1})^n = M D^n M^{-1}$. Taking the power of a diagonal matrix is easy, so this is often a nice way t... | 51 |
linear-algebra | Determine for with values of $a$ the matrix is diagonalizable over $\mathbb{R}$ | https://math.stackexchange.com/questions/1902987/determine-for-with-values-of-a-the-matrix-is-diagonalizable-over-mathbbr | <blockquote>
<p>Determine for which values of <span class="math-container">$a$</span> <span class="math-container">\begin{pmatrix} 4 & 0 & 0 \\ 4 & 4 & a \\ 4 & 4 & 4 \end{pmatrix}</span></p>
<p>The matrix is diagonalizable</p>
</blockquote>
<p>So we first look at the characteristic polynomial:<... | <p>You can do this without much computation.</p>
<p>First off, for $a=0$ the matrix is triangular (but not diagonal) with equal diagonal entries, therefore not diagonalisable.</p>
<p>Then assume $a\neq 0$. Now the matrix is block triangular, so its characteristic polynomial is the product of those of the diagonal blo... | 52 |
linear-algebra | $\mathbb{R}_{\le3}[X]$ is not a subspace of $\mathbb{R}_{\le4}[X]$ (polynomials in linear algebra) | https://math.stackexchange.com/questions/1904387/mathbbr-le3x-is-not-a-subspace-of-mathbbr-le4x-polynomials | <p>I'm sorry that this is probably a stupid question for this page, but I have no one to ask. I'm currently studying linear algebra by myself and I'm confused by this answer:</p>
<blockquote>
<p><span class="math-container">$V$</span> is not a subspace of <span class="math-container">$\mathbb{R}_{\le4}[X]$</span>, beca... | <p>If, as lisyarus asked, $\mathbb R_{\le n}[x]$ represents polynomials of at most degree $n$ then I somewhat disagree with that solution (and why subscripts as powers??). But here is why I think they <em>might</em> claim it. As vector spaces,</p>
<p>$$\mathbb R_{\le 4}[x] \simeq \mathbb R^5$$</p>
<p>Meaning they are... | 53 |
linear-algebra | Parallel vectors and rank of matrix | https://math.stackexchange.com/questions/2129639/parallel-vectors-and-rank-of-matrix | <blockquote>
<p>Suppose <span class="math-container">$v_1, v_2, v_3$</span> are (row) vectors in <span class="math-container">$\mathbb{R}^3$</span>, and they are parallel, then what you can say about the rank of the matrix:</p>
<p><span class="math-container">\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}</span></p>
<... | <p>If the vectors are parallel (that is, each $v_i$ is a constant multiple of each $v_j$) then the rank of the matrix is actually $\leq 1$ because the dimension of the row space (the span of the rows) is $\leq 1$. If some $v_i$ is non-zero then the rank will be one. If $v_1 = v_2 = v_3 = 0$ then the rank will be zero.<... | 54 |
linear-algebra | Rank of matrix containing NON parallel vectors | https://math.stackexchange.com/questions/2131053/rank-of-matrix-containing-non-parallel-vectors | <blockquote>
<blockquote>
<p>Suppose <span class="math-container">$v_1, v_2, v_3$</span> are (row) vectors in <span class="math-container">$\mathbb{R}^3$</span>, and they are <strong>not</strong> parallel, then what you can say about the rank of the matrix:</p>
</blockquote>
<p><span class="math-container">\begin{pmat... | 55 | |
linear-algebra | Dot product of projection and vector? | https://math.stackexchange.com/questions/2131202/dot-product-of-projection-and-vector | <blockquote>
<p>Suppose <span class="math-container">$P$</span> is a plane and <span class="math-container">$x$</span> is a vector (both in <span class="math-container">$\mathbb{R^3}$</span>), can we say that</p>
<p><span class="math-container">$$x \cdot \text{proj} _{P}x = 0$$</span></p>
</blockquote>
<p>For the dot p... | <p>Think about it. If p is some non-zero vector, the projection of u onto p is usually given by $$\frac{u\cdot{p}}{|p|^2}p$$ then,
$$
u\cdot{\frac{u\cdot{p}}{|p|^2}p}
$$
But we can see that $\frac{u\cdot{p}}{|p|^2}$is just a scalar that we can factor out of the dot product, thus we are left with
$$
\frac{u\cdot{p}}{|p... | 56 |
linear-algebra | A question about a proof that has to do with diagonal matrices | https://math.stackexchange.com/questions/2139992/a-question-about-a-proof-that-has-to-do-with-diagonal-matrices | <blockquote>
<p>Show that if <span class="math-container">$A$</span> is a diagonal matrix then orthogonal diagonalising matrix <span class="math-container">$Q = \text {Identity}.$</span></p>
<p>Proof: Let <span class="math-container">$A$</span> be a diagonal matrix and if <span class="math-container">$Q = I,$</span> th... | <p>A diagonalizable matrix $A$ is a matrix such that for invertible matrix $P$ and $D$ diagonal you have $A=PDP^{-1}$. Now for $A$ diagonal you can take $D=A$ and $P=I$ and this proves that $A$ is diagonalizable (as trivially expected).</p>
<p>Moreover orthogonally diagonalizable is when the matrix $P$ is orthogonal. ... | 57 |
linear-algebra | Structure of a mapping comes from the Codomain? | https://math.stackexchange.com/questions/2014698/structure-of-a-mapping-comes-from-the-codomain | <blockquote>
<p>Show that:</p>
<p>If <span class="math-container">$A$</span> is a non empty set and <span class="math-container">$R$</span> a ring, then <span class="math-container">$\operatorname{map}(A,R)$</span>, is a ring too, with the following operations:</p>
<p><span class="math-container">$f+g$</span> is define... | <p>It is not $f(x)+g(x)$, but $h=f+g$, with $h:A\rightarrow R$ such as $h(x)=f(x)+g(x)\in R$ by the properties of the ring for $x\in A$.</p>
| 58 |
linear-algebra | Prove that $X\oplus Y$ is a Dedekind cut | https://math.stackexchange.com/questions/2021755/prove-that-x-oplus-y-is-a-dedekind-cut | <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be a Dedekind cut. Now let <span class="math-container">$X\oplus Y=\{x+y\mid x\in X,y\in Y\}$</span>. Show that <span class="math-container">$X\oplus Y$</span> is again a Dedekind cut. i.e. it must fulfill the following condition... | <p>HINT: To show that $\Bbb Q\setminus(X\oplus Y)\ne\varnothing$, show that there are $r\in\Bbb Q\setminus X$ and $s\in\Bbb Q\setminus Y$ such that $r,s\ge 0$, and then show that $x+y<r+s$ for each $x\in X$ and $y\in Y$.</p>
<p>For (ii), suppose that $r\in\Bbb Q$ and $r<z\in X\oplus Y$. By definition there are $... | 59 |
linear-algebra | TRUE or FALSE? Eliminating z from x + 2y + 3z = 2, 3x + 2y + 3z = 6 and 2x + 3y = 5 gives x + 2y = 2 . | https://math.stackexchange.com/questions/2202276/true-or-false-eliminating-z-from-x-2y-3z-2-3x-2y-3z-6-and-2x-3y | <blockquote>
<p>Is the following statement true or false?</p>
<p>Eliminating <span class="math-container">$z$</span> from:<br />
<span class="math-container">$x + 2y + 3z = 2$</span>,<br />
<span class="math-container">$3x + 2y + 3z = 6$</span> and,<br />
<span class="math-container">$2x + 3y = 5$</span>;<br />
gives <... | <p>As you found value of $x$ after that,</p>
<p>$2x + 3y = 5$</p>
<p>Put value of $x=2$ in above equation,</p>
<p>$4+3y = 5$</p>
<p>$3y = 1$</p>
<p>$y=\frac 13$</p>
<p>Then you can put value of $x, y$ in resultant equation $x+2y=2$. Value of $x, y$ doesn't satisfy. So answer is false.</p>
| 60 |
linear-algebra | To find factor of a polynomial equation | https://math.stackexchange.com/questions/2205323/to-find-factor-of-a-polynomial-equation | <blockquote>
<p>One of the factors of <span class="math-container">$4x^2+y^2+14x-7y-4xy+12$</span> is equal to</p>
<ol>
<li><p><span class="math-container">$2x-y+4$</span></p>
</li>
<li><p><span class="math-container">$2x-y-3$</span></p>
</li>
<li><p><span class="math-container">$2x+y-4$</span></p>
</li>
<li><p><span c... | <p>You can do $2x-y=k$ and then</p>
<p>$$k^2+7k+12=(k+3)(k+4)$$</p>
<p>and then you get</p>
<p>$$(2x-y+3)(2x-y+4)$$</p>
| 61 |
linear-algebra | How do I show that an equation has a solution orthogonal to the nullspace? | https://math.stackexchange.com/questions/2041229/how-do-i-show-that-an-equation-has-a-solution-orthogonal-to-the-nullspace | <p>This was a question on a recent linear algebra midterm, and I had no idea where to start.</p>
<blockquote>
<p>Fix an <span class="math-container">$m\times n$</span> matrix <span class="math-container">$A$</span> and a column vector <span class="math-container">$\mathbf{b}$</span> of size <span class="math-container"... | <p>Start with a solution $x_0$, then write $x_0$ as
$$
x_0 = \hat{x_0} + z,
$$
where $\hat{x_0} = \mathrm{Proj}_{Nul A} (x_0)$ is the orthogonal projection of $x_0$ to $Nul A$. </p>
<p>Then for any $y\in Nul A$,
$$
y \cdot z = 0. $$
The vector $z$ is the solution you want. </p>
| 62 |
linear-algebra | Hermitian operators $\langle Av,v\rangle=0$ for all $v\in V$ then $A=0$ proof | https://math.stackexchange.com/questions/2383614/hermitian-operators-langle-av-v-rangle-0-for-all-v-in-v-then-a-0-proof | <blockquote>
<p>Theorem: Let <span class="math-container">$V$</span> be as before. If <span class="math-container">$A$</span> is an operator such that <span class="math-container">$\langle Av,v\rangle=0$</span> for all <span class="math-container">$v\in V$</span> then <span class="math-container">$A=0$</span>.</p>
</bl... | <p>(1) In matrix terms, and working
over the reals, we have the hypothesis $v^t A v=0$ for all vectors
$v$. This does <strong>not</strong> imply $v^t Aw=0$ for all $v$ and $w$. Consider
$\pmatrix{0&1\\-1&0}$. To get this implication we need symmetry. Here the
analogue of symmetry is the Hermitian condition.</p>... | 63 |
linear-algebra | Direct Sum Of Two Subspace Of $\mathbb{R}^{2\times 2}$ | https://math.stackexchange.com/questions/2567858/direct-sum-of-two-subspace-of-mathbbr2-times-2 | <blockquote>
<p>Let <span class="math-container">$V=\mathbb{R}^{2\times 2}$</span> and define the subspaces</p>
<p><span class="math-container">$$U=\left\{\begin{pmatrix} a&0\\ 0&d \end{pmatrix}: a,d\in \mathbb{R}\right\}$$</span></p>
<p><span class="math-container">$$W=\left\{\begin{pmatrix} a&b\\ c&d ... | <p>I'm not sure what you mean with "everything covered" - what exactly have you done?</p>
<p>Note that an element of $W$ is of the form (using $a+c=0$ and $b+d=0$):
$$\begin{pmatrix} -c &b\\ c& -b \end{pmatrix}$$
So a matrix in $U+W$ has the form:
$$\begin{pmatrix} a-c &b\\ c& d-b \end{pmatrix}$$
From ... | 64 |
linear-algebra | Intersection of two polynomial subspaces | https://math.stackexchange.com/questions/2570880/intersection-of-two-polynomial-subspaces | <p>I'm working through the following problem.</p>
<blockquote>
<p>Let <span class="math-container">$U = \{ p \in \mathbb{P}_3 : p(1) = 0 \}$</span> and <span class="math-container">$V = \{ p \in \mathbb{P}_3 : p(-1) = 0 \}$</span>. Here, <span class="math-container">$\mathbb{P}_3$</span> represents the space of polynom... | <p>Any polynomial in $\mathbb{P}_3$ is of the form $ax^3+bx^2+cx+d$ where $a,b,c,d\in\mathbb{R}$. If $p(1)=0$, then $a+b+c+d=0$, so we can write $d=-a-b-c$, and hence polynomials in $U$ are of the form $ax^3+bx^2+cx-a-b-c=a(x^3-1)+b(x^2-1)+c(x-1)$. The dimension of $U$ is therefore $3$, since $\{x^3-1,x^2-1,x-1\}$ is a... | 65 |
linear-algebra | Is $V$ a vector space over $R$ under these two operations? | https://math.stackexchange.com/questions/2097193/is-v-a-vector-space-over-r-under-these-two-operations | <p>Let <span class="math-container">$V=\{(a,b) : (a,b) \in \mathbb{R}\}$</span></p>
<p>Is <span class="math-container">$V$</span> a vector space over <span class="math-container">$\mathbb{R}$</span> under:</p>
<p><strong>Addition</strong>: <span class="math-container">$(a_1,a_2)+(b_1,b_2)=(a_1+b_2,a_2+b_1)$</span></p>... | 66 | |
linear-algebra | How to find The Roots of Orthogonal polynomial equation | https://math.stackexchange.com/questions/2102294/how-to-find-the-roots-of-orthogonal-polynomial-equation | <blockquote>
<p><span class="math-container">$P(z)$</span>, with roots <span class="math-container">$z_j$</span>'s for <span class="math-container">$0\leq j\leq a-1$</span>.</p>
<p><span class="math-container">$$P(z)=z^a+c_{a-1}z^{a-1}+\ldots+c_1z+c_0.$$</span></p>
</blockquote>
<p>I want to find the Roots of orthogona... | 67 | |
linear-algebra | Prove that the union of two bases in different subspaces is a basis for vector space | https://math.stackexchange.com/questions/2585101/prove-that-the-union-of-two-bases-in-different-subspaces-is-a-basis-for-vector-s | <p>This is a question from Finite-Dimensional Linear Algebra by Mark S. Gockenbach page 72 (Exercise 2.7.14). I hope to check my proof. Thank you.</p>
<blockquote>
<p>Let <span class="math-container">$V$</span> be an <span class="math-container">$n$</span>-dimensional vector space over a field <span class="math-contain... | <p>Since the vectors are in bases of different subspaces, they are non-zero and linearly independent. Therefore, their union is also a basis for the span of the new set.</p>
| 68 |
linear-algebra | Nontrivial solution for Ax=0 and Ax=b determine by pivot positions | https://math.stackexchange.com/questions/2230419/nontrivial-solution-for-ax-0-and-ax-b-determine-by-pivot-positions | <blockquote>
<p>A is a 3x2 matrix with two pivot positions.</p>
<p>(a) does the equation Ax=0 have a nontrivial solution</p>
</blockquote>
<p>Since the two pivot positions will create 0 in the entire column in which they are present and 1 in its own position in reduced row echelon form and the rightmost column is all 0... | <p>Your answer to (a) looks good. Question (b) can be asked alternately as $``$Can $\mathbb{R}^3$ be spanned by only two vectors in $\mathbb{R}^3$$"$? </p>
| 69 |
linear-algebra | $\langle Av_1,Av_2\rangle=ac\langle v_1,v_1\rangle+bd\langle v_2,v_2\rangle$? | https://math.stackexchange.com/questions/2398483/langle-av-1-av-2-rangle-ac-langle-v-1-v-1-ranglebd-langle-v-2-v-2-rangle | <blockquote>
<p>Define a rotation of <span class="math-container">$V$</span> to be a real unitary map <span class="math-container">$A$</span> of <span class="math-container">$V$</span> whose determinant is 1. Show that the matrix of <span class="math-container">$A$</span> relative to an orthogonal basis of <span class=... | <p>Note that the scalar product is bilinear and:
$$\langle Av_1,Av_2\rangle =\langle w_1,w_2 \rangle =\langle av_1+bv_2,cv_1+dv_2 \rangle =\langle av_1,cv_1+dv_2\rangle+\langle bv_2,cv_1+dv_2\rangle = ac\langle v_1,v_1\rangle +ad\langle v_1,v_2\rangle+bc\langle v_2,v_1\rangle +bd\langle v_2,v_2\rangle.$$
Since $\langle... | 70 |
linear-algebra | Determinant of a symmetric matrix a quadratic form proof | https://math.stackexchange.com/questions/2401585/determinant-of-a-symmetric-matrix-a-quadratic-form-proof | <blockquote>
<p>Let <span class="math-container">$V$</span> be the vector space over <span class="math-container">$\mathbb{R}$</span> of <span class="math-container">$2\times 2$</span> real symmetric matrices.
Show that the function <span class="math-container">$f$</span> on <span class="math-container">$V$</span> such... | <p>If you have a quadratic form over the real numbers, in finite dimension, the matrix is <strong>half the Hessian matrix of second partial derivatives</strong>. This time</p>
<p>$$
\left(
\begin{array}{rrr}
0 & 0 & \frac{1}{2} \\
0 & -1 & 0 \\
\frac{1}{2} & 0 & 0
\end{array}
\right)
$$</p>
<p... | 71 |
linear-algebra | Showing there is a unique basis $\{p_1, p_2, p_3\}$ of $P_2(R)$ with certain properties | https://math.stackexchange.com/questions/1949874/showing-there-is-a-unique-basis-p-1-p-2-p-3-of-p-2r-with-certain-pro | <blockquote>
<p><span class="math-container">$P_2(R)$</span> is the set of polynomials of degree two or lower.</p>
<p>Show that there is a unique basis <span class="math-container">$\{p_1, p_2, p_3\}$</span> of <span class="math-container">$P_2(R)$</span> with the property that <span class="math-container">$p_1(0) = 1,... | <p>You are asked to prove two things:</p>
<ul>
<li>Existance: there is such a basis, and</li>
<li>Uniqueness: there is at most one such basis.</li>
</ul>
<p>The proof you describe is only for proving uniqueness. Uniqueness is almost always easier to prove than existance.</p>
<p>To prove such a basis exists, you need... | 72 |
linear-algebra | Eigenvalues of Linear Operator given by Conjugation by an Invertible Matrix | https://math.stackexchange.com/questions/1955403/eigenvalues-of-linear-operator-given-by-conjugation-by-an-invertible-matrix | <p>I am working on a review problem for comp/qual studying and I cannot figure it out. The hint provided seems to give some intuition, but I don't see how it generalizes.</p>
<blockquote>
<p>Let <span class="math-container">$A \in GL(n,\mathbb{C})$</span> be an <span class="math-container">$n \times n$</span> invertib... | <p><strong>Hint:</strong> Let $E_{ij}$ be the matrix whose only non-zero entry is $e_{ij} = 1$. Then $E_{ij}$ is an "eigenvector" of $T$, and there are $n^2$ linearly independent such matrices.</p>
<p>Now, let $T_A(M) = A^{-1}MA$. Note that if $A = SDS^{-1}$, then
$$
T_A = (T_S)^{-1} \circ T_D \circ T_S
$$</p>
| 73 |
linear-algebra | U ∪W is a subspace => U is a subset of W or W is a subset of U (given that U and W are subspaces) | https://math.stackexchange.com/questions/2114139/u-%e2%88%aaw-is-a-subspace-u-is-a-subset-of-w-or-w-is-a-subset-of-u-given-that-u-and | <p>I wrote a proof of this (and yes, I proved the opposite direction, but I don't have a question about that portion), and I just want to get confirmation that I am not missing anything -- or advice on how to clean it up if it needs that.</p>
<p>Here's the proof:</p>
<blockquote>
<p>Suppose W is not a subset of U.</p>
... | 74 | |
linear-algebra | Minimum value of function $f(x, y)$ if $x$ and $y$ are real numbers and no other conditions are given | https://math.stackexchange.com/questions/2249488/minimum-value-of-function-fx-y-if-x-and-y-are-real-numbers-and-no-other | <p>So, I got this problem that's been bugging me. For a quick info, I'm a 12th grader in Indonesia. The problem was given by my teacher to evaluate my understanding on inequalities. Here is the problem:</p>
<blockquote>
<p>If <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are both r... | <p>Refering to the vector form of Schwarzy's inequality, I find that :
$$\sqrt {4+y^2} + \sqrt {(x-2)^2 +(2-y)^2} + \sqrt{(4-x)^2 + 1}$$
can be broken down to :
$$\sqrt {2^2+y^2} + \sqrt {(x-2)^2 +(2-y)^2} + \sqrt{(4-x)^2 + 1^2} \ge{} \sqrt{(2 + x - 2 + 4 -x)^2 + (y + 2 - y +1)^2}$$ which can then be simplified to : $$... | 75 |
linear-algebra | Proof for vectors added to a subspace are equal iff the difference is in the subspace | https://math.stackexchange.com/questions/2662417/proof-for-vectors-added-to-a-subspace-are-equal-iff-the-difference-is-in-the-sub | <blockquote>
<p>For any vector space <span class="math-container">$V$</span>, subset <span class="math-container">$S \subseteq V $</span>, and vector <span class="math-container">$\vec{v} \in V$</span>, we define the set
<span class="math-container">$$\vec{v}+S = \{\vec{v} + \vec{x} : \vec{x}\in S\} $$</span>
Prove tha... | <p>You are making it too complicated. If $a+W=b +W$, then there exist $u,v\in W$ such that $a+u=b+v$. So $a-b=v-u\in W$.</p>
<p>Conversely, if $a-b\in W$, there exists $w\in W$ with $a-b=w$. Then $a=b+w\in b+W$, so $a+W\in b+W$; and the reverse inclusion follows but reversing roles.</p>
| 76 |
linear-algebra | Triangulation of a matrix and the eigenvalues, right? | https://math.stackexchange.com/questions/2415203/triangulation-of-a-matrix-and-the-eigenvalues-right | <blockquote>
<p>Find the characteristic polynomial,eigenvalues, and bases for the eigenspaces of the following matrices.</p>
<p><span class="math-container">$\begin{bmatrix}4&0&1\\-2&1&0\\-2&0&1\end{bmatrix}$</span></p>
</blockquote>
<p>We know that <span class="math-container">$\det(tI-A)=0$</s... | <p>The eigenvalues of a matrix are not invariant under elementary row operations. They are however invariant under similarity transformation. </p>
| 77 |
linear-algebra | Distributing Basis Coordinates | https://math.stackexchange.com/questions/2671045/distributing-basis-coordinates | <blockquote>
<p>Let <span class="math-container">$V$</span> be a finite-dimensional vector space with (ordered) basis <span class="math-container">$\beta=(b_1,...,b_n)$</span>, and let <span class="math-container">$T:V\rightarrow V$</span> be a linear transformation. Let <span class="math-container">$B=[T]_\beta$</span... | <p>Yes $[Tv]_\beta = [T]_\beta[v]_\beta$. Note that for $v = a_1b_1 + \cdots + a_nb_n \in V$ we have:
$$
\begin{align}
[Tv]_\beta &= [a_1T(b_1) + \cdots + a_nT(b_n)]_\beta \\ &= a_1[T(b_1)]_\beta + \cdots + a_n[T(b_n)]_\beta
\end{align}
$$</p>
<p>The last equality follows because choosing a basis defines an i... | 78 |
linear-algebra | Are the set of vectors linearly dependent? | https://math.stackexchange.com/questions/2761116/are-the-set-of-vectors-linearly-dependent | <blockquote>
<p>Are the set of vectors linearly dependent?</p>
<ol>
<li><p><span class="math-container">$ \{ e^{x}, e^{-x}\} $</span> in <span class="math-container">$\mathcal{F} (\mathbb{R} ,\mathbb{R} )$</span></p>
</li>
<li><p><span class="math-container">$ \{ \frac{1}{x-1}, \frac{1}{x + 1} \} $</span> in <span clas... | <p>Guide:</p>
<p>Try to substitute some value of $x$, for example, in the first example, we can let $x=0$ and we obtain $a+b=0$, try to obtain another condition for $a$ and $b$ by letting $x$ equal to another value and then you can solve for $a$ and $b$.</p>
<blockquote class="spoiler">
<p> yes, they are linearly i... | 79 |
linear-algebra | Theorems on 1-to-1 and onto linear functions | https://math.stackexchange.com/questions/2442512/theorems-on-1-to-1-and-onto-linear-functions | <p>I'm given two theorems:</p>
<blockquote>
<p>Theorem (1)</p>
<p>Following statements are equivalent:</p>
<p><span class="math-container">$(i)$</span> <span class="math-container">$L : V \to U$</span> where <span class="math-container">$V$</span> and <span class="math-container">$U$</span> are vector spaces, is one to... | 80 | |
linear-algebra | Orthogonal projection of a point into $x+y+z=0$ plane ex. | https://math.stackexchange.com/questions/2326429/orthogonal-projection-of-a-point-into-xyz-0-plane-ex | <blockquote>
<p>Let <span class="math-container">$T:\mathbb{R}^3\to W$</span> be the orthogonal projection of <span class="math-container">$\mathbb{R}^3$</span> onto the plane <span class="math-container">$W$</span> having the equation <span class="math-container">$x+y+z=0$</span>.</p>
<p>(a)Find <span class="math-cont... | <p>Let $P(a,b,c)\in\mathbb{R}^3$. The line which passes through $P$ and is orthogonal to $W$ is </p>
<p>$$\vec{r}=(a,b,c)+t(1,1,1)=(a+t,b+t,c+t)$$</p>
<p>At the intersection of the line and $W$ (which is $T(P)$),</p>
<p>\begin{align}
a+t+b+t+c+t&=0\\
t&=\frac{-1}{3}(a+b+c)
\end{align}</p>
<p>So, $$T(a,b,c)=... | 81 |
linear-algebra | $\mathscr{M}_{\beta´}^{\beta}(id)$ in $\mathbb{R}^3$ | https://math.stackexchange.com/questions/2335624/mathscrm-beta%c2%b4-betaid-in-mathbbr3 | <blockquote>
<p>In each one of the following cases, find <span class="math-container">$\mathscr{M}_{\beta´}^{\beta}(id)$</span>. >The vector space in each case is <span class="math-container">$\mathbb{R}^3$</span>.</p>
<p>a) <span class="math-container">$\beta=\{(1,1,0),(-1,1,1),(0,1,2)\}\\\beta´={(2,1,1),(0,0,1),(-... | <p>Here's a first step:</p>
<p>Let $\beta = \{v_1,v_2,v_3\}$ and $\beta' = \{w_1,w_2,w_3\}$. We note that
$$
\operatorname{id}(v_1) = v_1 = \frac 23 w_1 + (-1)w_2 + \frac 13 w_3
$$
As such, we will find that
$$
\mathcal M^{\beta}_{\beta'}(\operatorname{id}) = \pmatrix{2/3 &?&?\\-1&?&?\\1/3&?&?... | 82 |
linear-algebra | Checking For Orthogonality | https://math.stackexchange.com/questions/2449427/checking-for-orthogonality | <blockquote>
<p>Let <span class="math-container">$C[-1,1]$</span> and <span class="math-container">$f:[-1,1]\to \mathbb{C}$</span> with the inner product <span class="math-container">$\langle f,g\rangle=\int_{-1}^{1}f(x)\overline{g(x)}dx$</span></p>
<p>Prove: <span class="math-container">$P_{0}=1, P_{1}=x, P_{2}=1-3x^2... | <p>Actually, it all depends on what you admit first.</p>
<p>Recall that an inner product must verify $\langle v,v \rangle = 0 \Leftrightarrow v = 0$ for all $v$ in the vector space.</p>
<p>So, if you admit the application you defined here is an inner product, then you know that for any function $f$ different from $0$... | 83 |
linear-algebra | Showing Positive Definiteness In Inner Product | https://math.stackexchange.com/questions/2449525/showing-positive-definiteness-in-inner-product | <blockquote>
<p>Let <span class="math-container">$\mathcal{P}_2$</span> be the space of all polynomials of degree less or equal to <span class="math-container">$2$</span> for all <span class="math-container">$f,g\in \mathcal{P}_2$</span> we define:</p>
<p><span class="math-container">$$\langle f,g \rangle=\int_0^\infty... | <p>We know that $e^{-x}> 0$ and $(f(x))^2\geq 0$. Thus, $\displaystyle\int_{0}^{\infty} (f(x))^2 e^{-x}\geq 0$ and this function is the product of continuous functions, therefore, is continuous. Then, $\displaystyle\int_{0}^{\infty} (f(x))^2 e^{-x}=0$ if and only if $(f(x))^2 e^{-x}=0$ if and only if $(f(x))^2=0$ if... | 84 |
linear-algebra | Is this map linear? | https://math.stackexchange.com/questions/2486437/is-this-map-linear | <p>Is this map linear?</p>
<h2><span class="math-container">$T(x_1,x_2)=(x_1+2x_2+3,x_2+2x_1,3x_1)$</span></h2>
<p>Thank you very much! I thought it is not linear because there is a constant, which causes <span class="math-container">$T(v)+T(u)$</span> not to equal to <span class="math-container">$T(v+u)$</span>.</p>
| <p>Let $u=(x_1,y_1)$ and $v=(x_2,y_2)$.</p>
<p>For a transformation to be linear, two conditions must hold: $T(u+v)=T(u)+T(v)$ and $T(cu)=cT(u)$ for $c \in \mathbb{R}$ (a quicker, one step test is to check $T(c_1u+c_2v)=c_1T(u)+c_2T(v)$)</p>
<p>Consider $T(cu)$</p>
<p>$T(cu)=T((cx_1,cy_1))=(cx_1+2cy_1+3,cy_1+2cx_1,3... | 85 |
linear-algebra | $ix+y-z=0\\iy+z=0$ basis | https://math.stackexchange.com/questions/2361760/ixy-z-0-iyz-0-basis | <blockquote>
<p>Find the dimension over <span class="math-container">$\mathbb{C}$</span> of the space of solutions of the following systems of equations. Also find a basis for this space of solutions.</p>
<p><span class="math-container">$ix+y-z=0\\iy+z=0$</span></p>
</blockquote>
<p>Using the formula <span class="math-... | <p>Since the so called "space of solutions" has the dimension $1$, any element in the space forms a basis for the space because the solutions of this system of equations represents a line, and any point in a line can be written as a linear combination of another line.</p>
<p><strong>Edit:</strong></p>
<p>From the equ... | 86 |
linear-algebra | Derivatives $f:\mathbb{R}^n\to\mathbb{R}$ quadratic form? | https://math.stackexchange.com/questions/2374055/derivatives-f-mathbbrn-to-mathbbr-quadratic-form | <p>Let <span class="math-container">$f:\mathbb{R}^n\to\mathbb{R}$</span> be a twice continuously differentiable function such that <span class="math-container">$f(tX)=t^2f(X)$</span> for all <span class="math-container">$X\in\mathbb{R}$</span>. Show that <span class="math-container">$f$</span> is a quadratic form.(You ... | <p>$f(tx)=t^2f(x)$</p>
<p>Let's take derivatives with respect to $t$. We denote by $D_i$ the derivative with respect to $x_i$. </p>
<blockquote>
<p>On the right-hand side the derivative is $2tf(x)$ since $f(x)$ is just a constant with respect to $t$. On the left-hand side we need to apply the chain rule in several ... | 87 |
linear-algebra | Let $T : \mathbb R^m \to \mathbb R^n$ be a linear transformation, prove that... | https://math.stackexchange.com/questions/1144193/let-t-mathbb-rm-to-mathbb-rn-be-a-linear-transformation-prove-that | <blockquote>
<p>Let <span class="math-container">$T : \mathbb R^m \to \mathbb R^n$</span> be a linear transformation.</p>
<ol>
<li>Prove that <span class="math-container">$T$</span> is injective if and only if for every linearly independent set <span class="math-container">$\{\overrightarrow v_1,\ldots,\overrightarrow ... | <p><strong>Hint</strong>: take $v_1,\dots,v_k$ to be a set of linearly independent vectors. Suppose that $c_1,\dots,c_k$ are scalars such that
$$
c_1T(v_1) +\cdots+c_k T(v_k)= 0
$$
If $T$ is injective, show that each $c_i$ must be zero. </p>
<p>If $T$ is not injective, show that we can find a non-zero $v$ so that $T(... | 88 |
linear-algebra | Show equality of two krylov spaces | https://math.stackexchange.com/questions/3721106/show-equality-of-two-krylov-spaces | <p>A bijective funktion f:V->V and a m-krylov-space K_m(f,v)=spann{v,f(v),..,f^(m-1)(v)} are given.</p>
<p>We have to show f(K_m(f,v))=K_m(f,v), so practically spann{v,f(v),..,f^(m-1)(v)}=spann{f(v),f^2(v),..,f^m(v)} if i understand it correctly.
I don't quite see how i can use the bijectivity of f here</p>
| 89 |
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