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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Powerful applications of linear algebra?

I'd like to see some neat, elegant applications of linear algebra. I'm a undergrad but I don't want to prevent people from posting things just because I won't understand them, but if it's undergrad ...
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2 views

Find equations such that the solution space is the Image of T

Suppose T is a linear transformation such that (x1, x2, x3) -> (3x1 + 4x2 + 2x3, x1 + 2x2, 2x1 + x2 + 3x3, -x1 + 5x2 - 7x3) Find a homogeneous system of equation such that Image(T) = Solution space ...
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1answer
7 views

Finding a transformation matrix given a basis of matrices

I am looking for the transformation matrix of T using the basis E and the linear transformation listed below. I'm not confident that the way I am solving the problem is the correct way and if the ...
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5answers
67 views

Meaning of $\simeq$ symbol.

What does $\simeq$ mean and how is it used? I couldn't search online about it because I really don't know what it is!
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12 views

Proving rank inequalities of linear maps

I am given linear maps S and T and need to prove the four things. For the first two examples should I consider the subsets and dimensions of the vector spaces and then use the rank-nullity theorem? ...
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0answers
10 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 ...
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0answers
19 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 ...
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22 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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22 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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1answer
18 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 ...
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0answers
20 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
32 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
16 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
12 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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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 $D$ ...
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0answers
11 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 ...
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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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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 ...
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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 ...
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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
17 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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0answers
21 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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0answers
22 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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0answers
5 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 ...
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1answer
22 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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2answers
32 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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23 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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0answers
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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
16 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
21 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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0answers
26 views

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
80 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
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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_{++} ...
2
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1answer
27 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
16 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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0answers
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
44 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
27 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
88 views

Possible method to prove 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
66 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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0answers
12 views

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 ...
0
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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 ...
0
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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 ...