Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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4
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1answer
482 views

Rank of a rectangular Vandermonde Matrix to which weighted columns are added

A Vandermonde matrix: $\left(\begin{array}{ccc} 1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\ 1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\ \vdots & \vdots & \ddots & ...
0
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0answers
3 views

Derivation of the Binomial Inverse Theorem

Can anyone suggest a reference that details the derivation of the Binomial Inverse Theorem -- specifically, the first equation in https://en.wikipedia.org/wiki/Binomial_inverse_theorem and not the ...
3
votes
1answer
13 views

On the product of involution matrices

Let $F$ be a field and let $A\in M_n(F)$ be a matrix with $det(A) = \pm 1 $. How can I show that $A$ is a product of involutions ? Of course the converse is true and clear. By involution I mean a ...
7
votes
1answer
386 views

Eigenvalues of a tridiagonal trigonometric matrix

Let $D$ be the diagonal matrix w/alternating in sign diagonal entries: $$D_{kk}=(-1)^{k+1}\tan(\frac{k\pi}{2n+1}),$$ where $k=1,2,\dots n\in N$, and let $B$ be the $n$ by $n$ square $(0,1)$-matrix ...
0
votes
1answer
10 views

How to find which variable impacts the answer the most in this equation?

In this equation, if two of the variables are held constant, which variable will bring out the maximum positive change in the answer? I tried doing this in excel, but I'm having trouble figuring out ...
0
votes
0answers
11 views

Notation fo the reverse identity matrix

I'm wondering if there's a canonical notation for the reverse identity matrix, i.e. $$ ?=\left(\begin{array}{ccccc} 0 & 0 & 0 & 0 & 1\\ 0& 0 & 0 & 1 & 0\\ 0 & 0 ...
0
votes
0answers
16 views

Matrix acting on a tensor product

What does it mean for a matrix to act on a tensor product? I think there is a disconnect between vocabulary I am using and vocabulary the professor is using. Specifically, I have a $2 \times 2$ matrix ...
-1
votes
0answers
20 views

Formula for finding integer solutions to Ax=b?

How can I generate nontrivial (a : Integer, b : Integer) so that: $$ \begin{pmatrix}a&0&0 \\ 0 & a & 0 \\ 0 & 0 & a\end{pmatrix} \begin{pmatrix}b \\ b \\ b\end{pmatrix} = ...
4
votes
1answer
59 views
+50

Find (linear) transformation matrix using the fact that the diagonals of a parallelogram bisect each other.

This is the first time I post something on this website. I'm on this question already for hours. I'm clearly not asking the community to do my homework, I'm hoping someone can explain me how I should ...
7
votes
0answers
165 views

Inverse (finite group) isomorphism of a certain form exists

I have been working something in group theory for a long time and I have everything worked out but this one problem. I have reduced that problem to a conjecture. It takes some work to set it up, but I ...
0
votes
0answers
17 views

Characteristics of $\vec u$ in equation of the reflection of $\vec x$ about the line $N$

The linear transformation of the reflection of $\vec x$ about line $N$ is $$\vec x = 2 \text{proj}_N(\vec x)\vec x - \vec x= 2(\vec x \cdot \vec u) \vec x- \vec x.$$ Is the unit vector, $\vec u$, ...
1
vote
1answer
19 views

Get coordinates in a scaled and translated canvas

I'm drawing an image to a scaled and translated HTML5 canvas. It's zoomed to a certain point on the image to be more specific, so only a smaller partition is visible. In this smaller part, I want to ...
6
votes
1answer
42 views

Differentiation as Rotation

I am trying to make a connection between linear algebra and the Fourier transform. Functions form a vector space and differentiation is an operator. Fourier transforming a function from what i ...
0
votes
1answer
12 views

Meaning of the phrase “Line $Y$ spanned by $\vec x$” and “Plane $D$ spanned by $\vec x$, $\vec y$, and $\vec z$”

If I say the Line $Y$ spanned by $\vec x$ in $\mathbb{R}^2$ = $\begin{bmatrix}3 \\2\end{bmatrix}$, then do I mean that $\vec x$ is parallel or perpendicular to Line $Y$? If I say the Plane $D$ ...
0
votes
0answers
87 views

Determinant of this matrix? [on hold]

How can I find the determinant of this matrix? I replaced each row starting from the thrid with the difference of the one before and it. In this way i transformed it into an almost diagonal matrix ...
0
votes
2answers
30 views

Find 2x2 matrix such that its inverse equals its transpose

Find some matrix $B\in GL_2 (\mathbb{R})$ such that $B^{-1} = B^T$ and $B \neq I$ What I tried: I tried to create a simultaneous equation i.e. if B = $\begin{bmatrix} a&b\\c & ...
0
votes
1answer
14 views

Is there any Eigen value decomposition, which can be warm-started?

I have a Matrix, $A$ which is positive semidefinite. No consider, $B=A+\Delta$. I have Eigen decomposition of $A$ and $\Delta_{ij}<= \epsilon_1$, $\Vert \Delta \Vert_F <= \epsilon_2$. Is there ...
0
votes
1answer
29 views

Finding the orthogonal basis, picture included!

I decided to share a picture of what I have so far. I am not sure if I did it correctly and sorry if it is not readable. Ask me if anything is unclear. In the exercise I am basically just asked to ...
0
votes
1answer
7 views

Find unit vector perpendicular to x-z,x-y, and y-z plane

I'm guessing that the unit vector perpendicular to the x-z plane is $\begin{bmatrix}1\\0\\1\end{bmatrix}$ I'm guessing that the unit vector perpendicular to the x-y plane is ...
0
votes
2answers
15 views

Prove constant times invertible matrix is also invertible

Let $B\in GL_n(\mathbb{R})$ and $\beta \in \mathbb{R}$ with $\beta \neq 0$. Show $\beta B \in GL_n(\mathbb{R})$ What I tried: I know it intuitively makes sense that this would be the case, but I ...
0
votes
1answer
11 views

Geometric and Algebric multiplicity of a Matrix

I'd like to proof that this matrix$$ A=\left(\begin{array}{ccccc} 0 & 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 3 & 0 & 0\\ 0 & 4 & 2 & 3 ...
1
vote
1answer
11 views

Why is $0$ an eigen value of $L_G$?

I am learning Spectral Graph Theory. If the Laplacian Matrix of a graph $G=(V,E)$ is defined by $(a_{ij})=-1 ;(i,j)\in E, d_i ; i=j$ and $0$ otherwise then how does it follow that $0$ is an ...
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0answers
9 views

counting the number of invertible matrices with entries in a specified field

Count the number of $n\times n$ invertible matrices modulo $26$. So far I am aware that a matrix is invertible if and only if its columns are linearly independent. I am also aware that the number of ...
1
vote
0answers
27 views

finding eigenvalues and eigenvectors

Find the eigenvalues and eigenvectors \begin{pmatrix} -7 & 0 & -8 \\ 2 & 1 & 2 \\ 6 & 0 & 7 \end{pmatrix} $\begin{bmatrix} -7-x & 0 & -8 \\ 2 & 1-x & 2 ...
0
votes
1answer
10 views

How to find scalar multiples that would make sum of matrices the zero matrix

What are all the possible values of $c_1$,$c_2$,$c_3$ $\in$ R such that $c_1$$\begin{bmatrix} 1&0\\ -1&0 \end{bmatrix} $ + $c_2$$\begin{bmatrix} 2&1\\ -2&2\end{bmatrix} $ ...
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vote
3answers
3k views

A vector that is orthogonal to the null space must be in the row space

Simple question. We know from the fundamental theorem of linear algebra that the nullspace of a matrix is the orthogonal complement of its row space. I can write this as: Let $M$ be a matrix. The ...
0
votes
0answers
21 views

Let $U$ and $V$ be vector spaces of dimensions $n$ and $m$ over $K$. Find the dimension and describe a basis of $\operatorname{Hom}_K(U,V)$ [duplicate]

I am given vectors spaces $U$ and $V$ of dimensions $n$ and $m$ over $K$. How can I find the dimension and basis of $\operatorname{Hom}_K(U,V)$ ?
3
votes
3answers
39 views

$A$ has more columns than rows and has full row rank, show there exist infinitely many $B$ s.t. $AB=I$

If A $\in M_{m\times n}(R)$ such that $n>m$. Prove that if $\text{rank} (A) = m$ then there are infinitely many matrices $B \in \ M_{n\times m} (R)$ such that $ AB = I_m$ So the question is ...
0
votes
1answer
18 views

Let $T$ be a defined linear map. Write down the matrix of $T$ using the standard basis of $\mathbb{R}^2$ and secondly using the basis $(1,-1),(0,-2)$. [on hold]

So I am given a linear map $T$ which is specifically defined. I have to find a matrix of $T$ using the standard basis and then using the given basis. I am not sure how to approach this problem?
1
vote
1answer
20 views

exponentiating a matrix and sum of elements

$$ M= \begin{bmatrix} 1&1&0\\0&1&1\\0&0&1\\ \end{bmatrix} $$ Then the sum of all entries of $e^{M}$ i just don't know how to calculate this sum as this would be an infinite ...
1
vote
1answer
24 views

Understanding Dual of $l_1$ norm

I am new in vector and Matrix Calculus. I was studying different norm and dual norm of vectors. I found $L_1$ norm as defined as the sum of absolute value of the points in the vector. The dual of that ...
0
votes
0answers
19 views

Find vector in $\mathbb{R}^2$ parallel to line and vector in $\mathbb{R}^3$ parallel to plane in $\mathbb{R}^3$

In $\mathbb{R}^2$ Given the line $f(x)=mx+b$, how do I find the vector parallel to it? For example, if I have the line $f(x)=4x+3$ which in in the form $f(x)=mx+b$, then is one of the vectors ...
1
vote
2answers
16 views

Cancellation law for invertible matricies

Show that the cancellation law holds for invertible matrices. i.e. if $A \in GL_n(R), B, C \in M_{n×m}(\mathbb{R})$ and $AB = AC$, then $B = C$. What I tried: I know that I can prove this by ...
1
vote
1answer
1k views

Range of A and null space of the transpose of A

So I'm complete stuck with something. I know it the following statements are true (or at least the seem to be from the results that I got from messing around with it a bit on MATLAB), but I don't ...
0
votes
0answers
3 views

Unique least squares solution for bounded variables of overdetermined rank-deficient linear system?

I am trying to solve an overdetermined linear system $A x = b$ where $A \in \mathbb{R}^{m \times n}$ $m > n $ $rank(A)<n$ $0 \leq x \leq u $ (all entries are bounded) $A, b \geq 0 $ (all ...
22
votes
6answers
20k views

Physical Meaning of Null Space of a Matrix

What is an intuitive meaning of the null space of a matrix? Why is it useful? I'm not looking for textbook definitions... my textbook gives me the definition, but I just don't "get" it. E.g.: I ...
0
votes
4answers
38 views

Derivative of matrices product

Find the derivative of the following matrix $ f(X) = a^TXb, $ where $ a,b ∈ R^n $ and X is an n×n matrix. Please give me some serious hint!
2
votes
0answers
22 views

A simple way to solve that inner product and functional question

Let $V$ be the polynomials space over $R$ of degree less than 3 with inner product $$\langle f,g\rangle= \int^{1}_{0} f(x)g(x) dx $$ if $x \in R$, compute $g_{x}$ such that $\langle f,g_{x}\rangle ...
1
vote
1answer
22 views

Linear Algebra matrices question.

Let $A,B$ be 2 square matrices of the same size. And the following holds true $AB=A+B$ How do I prove that $(I-B)$ and $(I-A)$ are invertible
0
votes
1answer
397 views

Connection between linear independence, non-/trivial and x solutions

I am having a hard time remembering which goes hand in hand with what. The math questions I get always include words like trivial etc. 1 solution no solution infinite amount of solutions And then ...
2
votes
3answers
78 views

Is there a geometric meaning associated with the condition “dot product equals $1$?”

Consider $x,y \in \mathbb{R}^n$. Then the condition $x \bullet y = 0$ is easy to understand; it just means that $x$ and $y$ are orthogonal. Question. Does the condition $x \bullet y = 1$ have an ...
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0answers
12 views

Gramian Matrix Eigenvalues--Stronger Statement than Non-Negative

I'm struggling to find conditions under which this holds: $AA^T - B \succeq 0\,.$ If it helps, A is not necessarily square and $A_{ij} \in \{-1, 0,1\}$. B is diagonal and I would like to find ...
0
votes
1answer
483 views

Construct a matrix given basis for column space and basis for row space [GStrang P193 3.6.22]

Construct $A = \mathbf{c_1{r_1}^T + c_2{r_2}^T}$ whose column space has basis $(1, 2, 4), (2, 2, 1)$ and whose row space has basis $(1, 0), (1, 1)$. Answer: $A = \begin{bmatrix} ...
3
votes
0answers
51 views
+50

Proving Holder's inequality for Schatten norms

Sticking to the finite dimensional case, Holder's inequality for Schatten norms is given by $$\left\|AB\right\|_{S^1}\leq\left\|A\right\|_{S^p}\left\|B\right\|_{S^q}$$ for $A,B$ $n\times n$ ...
3
votes
1answer
20 views

Infinite dimensional topological vectorspaces with dense finite dimensional subspaces

Consider $\mathbb R$ as a $\mathbb Q$ vector space. Using the usual metric on $\mathbb R$, we find: $\mathbb Q \subset \mathbb R$ is dense and one dimensional (indeed every non-zero subspace appears ...
0
votes
0answers
14 views

why dual of $l_1$ norm is $l_\infty$ and vice versa

This might be a very dumb question but I am having a hard time to understand why dual of $l_1$ norm is $l_\infty$ and vice versa. The dual of a norm is denoted $\lVert\cdot\rVert_*$, defined as $$ ...
0
votes
1answer
16 views

Eigenvalues of positive linear combination of p.d. matrices

I want to prove a property on the eigenvalues of a positive linear combination of p.d. matrices. I have the following: $$ z \in \mathbb R^m_{++} $$ $$ A(z) = \Sigma z_i A_i $$ $$A_i \in S^n_{++} ...
3
votes
2answers
87 views

What are some usual norms for matrices?

I am familiar with norms on vectors and functions, but do there exist norms for spaces of matrices i.e. $A$ some $n \times m$ matrix? If so, that would that imply matrices also form some sort of ...
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0answers
50 views

XOR binary matrix multiplication $AX=B$? [on hold]

Let $A$, $B$, and $X$ be binary matrices (in F2 ) {0,1} positive values, where $A$ and $B$ are of size $n \times m$ with $n > m $ (more equations than variables). $X$ is an $m \times m$ matrix. ...
2
votes
1answer
25 views

Dual map is zero if and only if map is zero

A problem from Linear Algebra Done Right (Third Ed): Suppose $W$ is finite dimensional and $T \in \mathcal{L}(V, W)$. Prove that $T=0$ if and only if its dual $T' \in \mathcal{L}(W', V')=0$. I am ...