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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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 ...
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9 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 ...
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12 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 ...
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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} = ...
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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 ...
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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, $\vec u$, ...
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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 & ...
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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 ...
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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 ...
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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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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 $D$ ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\operatorname{Hom}_K(U,V)$ ?
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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 ...
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1answer
17 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?
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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
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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 ...
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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 ...
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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 $$ ...
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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 ...
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Looking for easygoing, well-motivated introductions to matrix norms.

I find all the various matrix norms very hard to navigate, probably because I don't know what they're used for. Question. What are some easygoing, well-motivated introductions to matrix norms? ...
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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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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_{++} ...
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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 ...
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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 ...
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30 views

Prove that $\operatorname{adj}(A) = \frac{1}{2}[(\operatorname{tr} A)^2 - \operatorname{tr}(A^2)]I_3 - [\operatorname{tr}(A)]A + A^2 .$ [duplicate]

Let $A$ be a square matrix of order $3$. Prove that $$\operatorname{adj}(A) = \frac{1}{2}[(\operatorname{tr} A)^2 - \operatorname{tr}(A^2)]I_3 - [\operatorname{tr}(A)]A + A^2 $$ where ...
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2answers
43 views

Calculating the determinant as a product without making any calculations

My problem is on the specific determinant. $$\det \begin{pmatrix} na_1+b_1 & na_2+b_2 & na_3+b_3 \\ nb_1+c_1 & nb_2+c_2 & nb_3+c_3 \\ nc_1+a_1 & nc_2+a_2 & nc_3+a_3 ...
2
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1answer
18 views

Understanding the definition of the direct sum of subspaces of a vector space

I have a question regarding the definition of direct sum of a vector space in relation to subspaces. Definition: A vector space $V$ is called the direct sum of $W_1$ and $W_2$ if $W_1$ and $W_2$ ...
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4answers
26 views

Show that a matrix with (I) a row of zeros and (II) a column of zeros cannot be invertible (respectively)

Show that a matrix with a row of zeros cannot be invertible. Show that a matrix with a column of zeros cannot be invertible. What I tried: I tried to show that a matrix $A \in M_n (\mathbb{R})$ such ...
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1answer
46 views

Show that $(I − Q)^{−1} $= $Q^2 + Q+ I$.

Consider $Q\in M_n (\mathbb{R})$ Assume that $Q^3 = [0] $ show that $ (I − Q)^{−1} = Q^2 + Q + I$. What I tried: I tried to use $(I-Q)(I-Q)^{-1} = I$ and use that to manipulate the left side of the ...
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1answer
71 views

Possible proof of infinite twin prime conjecture

I have an idea for proving the infinite twin prime conjecture that would set up a correspondence between primes. Since they've been proven infinite, twin primes would be shown infinite. Here it is: ...
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2answers
55 views

Use matrix algebra to show $A(B^{-1}(A+B)A^{-1})B = A+B$

I've got a super simple linear algebra question for an intro college course I can't seem to figure out. Using matrix algebra and matrix identities, show that: $$ A(B^{-1}(A+B)A^{-1})B = A+B $$ ...
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1answer
18 views

If $A$ Is an Upper Triangle Matrix, the Adjoint Is Also Upper Triangular

I already proved it, but it was really laborious. I am wondering if any one has a shorter proof? Write $A = [a_{ik}]$ and let $\overline{A}_{rs} = [c_{ik}]$ denote the minor with row $r$ and column ...
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2answers
65 views

Applications of Linear Algebra in software engineering.

I'm a software engineering and mathematics student, I was searching for disciplines of mathematics that would go well with my engineering degree, and found a lot of people recommended that software ...
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Proving the basis of an eigenspace is not the same for a matrix and its transpose

With the information given in #18, prove that $A$ and $A^T$ need not have the same eigenspaces (I would use a 2x2 matrix, as #18 posits). Clarification: DO NOT solve #18. Using the information in ...
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1answer
15 views

2-Norm of Non-Square Matrices

So, the 2-norm of an m x n matrix for m=>n is defined by the max singular value/square of the max eigenvalue. But, if it's not square, and you're only given a matrix A (no x-vector), what do you do if ...
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1answer
23 views

Find the distance from the point B to a line l.

So we have the point B = (2, 2) and the equation [x,y] = [-1, 2] + t[1, -1]. I know the first thing we need to do is calculate a point on the line, P. I did this by choosing a value for t, and then ...
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1answer
11 views

Find a transformation matrix between designated points in a photo and on a map

I took a photo of Athens from higher ground, and wrote a small in-browser app that allows me to set points on both the photo and on google maps. Screenshot below: (large version here) I want to ...
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34 views

Writing an expression in terms of vectorization operator vec(X)

I am new with Vectorization and Kronecker products. I need to write the following scaler value in terms of $\mathrm{vec}\left(\mathbf{X}\right)$ not $\mathbf{X}$: ...
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1answer
28 views

Linear transformation representation proof

I am wanting for someone to go over what I have and possibly correct my mistakes. Or any comments on the techniques, etc. I want to prove that if $V$ and $W$ are vector spaces over some field $F$, ...
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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 ...
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1answer
36 views

Gram-Schmidt Process to find an orthonormal basis for a matrix

By using the Gram-Schmidt Process find an orthonormal basis for the column space of the matrix: $$A=\begin{pmatrix}0 & -3 & 1 \\ 1 & 0 & 1 \\ 1 & -3 & ...
4
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1answer
36 views

Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$

Suppose that $M$ is symmetric idempotent $n\times n$ and has rank $n-k$. Suppose that $A$ is $n\times n$ and positive definite. Let $0<\nu_1\leq\nu_2\leq\ldots\nu_{n-k}$ be the nonzero ...