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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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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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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Let U and V be vector spaces of dimensions n and m over K. Find the dimension and describe a basis of Homk(U,V)

I am given vectors spaces U and V of dimensions n and m over K. How can I find the dimension and basis of Homk(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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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 $(1,-1),(0,-2)$.

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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21 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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20 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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6 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 ...
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12 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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74 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
23 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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11 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 ...
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1answer
17 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
24 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
45 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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68 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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53 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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17 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
62 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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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
14 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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22 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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28 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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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 & ...
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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 ...
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35 views

Under What Intervals Is A Matrix Positive Definite, Positive Semi-Definite, Indefinite, Negative Definite and Negative Semi-Definite?

Suppose we have a matrix which represents a quadratic form. $$ \begin{matrix} a & -a & -3a \\ -a & 2a & 2a \\ -3a & 2a & (9a+2) \\ ...
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Find a basis for the column space - why not reduce to RREF first?

Related to Understanding how to find a basis for the row space/column space of some matrix A. . When asked to find the basis for the column space of a matrix, can I first reduce to RREF, and then use ...
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1answer
106 views

Proving adjugate of $A$ for $3 \times 3$

From Wikipedia's article on adjugate matrix, Cayley–Hamilton theorem allows the adjugate of $A$ to be represented in terms of traces and powers of $A$. For the $3 \times 3$ case: ...
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Find the homothetic transformation

In $\mathbb{R^3}$: Find the homothety $\Phi$, such that the following transformations are possible: $$\Phi(P)=\Phi(1,0,-1)= (2,5,0)$$ and $$\Phi(Q)=\Phi(0,1,2)= (0,5,2)$$
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1answer
18 views

Matrix rank and number of linearly independent rows

I wanted to check if I understand this correctly, or maybe it can be explained in a simpler way: why is matrix rank equal to the number of linearly independent rows? The simplest proof I can come up ...
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2answers
49 views

Find $\det(A)$ of Matrix and condition on a and b

Let $$ A=\begin{bmatrix} a & b & 1 \\ b & 1 & b \\ 1 & a & a \\ \end{bmatrix} $$ Find $\det(A)$ in terms of $a$ and $b$, and write down ...
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$A$ 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 ...
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1answer
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Is a metric's form determined by its signature?

Suppose that we define a 4-dimensional vector space over the real field with a metric with signature (3, 1). Is the scalar product map determined only with this information? For example: a Minowsky ...
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55 views

Prove that $A+2I$ is invertible [duplicate]

Given $A$ is a square matrix such that $A^{3} = 2I$ Prove that $A+2I$ is invertible and find its inverse. How do I prove that $A+2I$ is invertible? For proving $A-I$ is invertible, I use ...
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1answer
25 views

Simpler way to show $v$ must be zero?

Let $x$ and $y$ be linearly independent vectors in $\mathbf{R}^2$. If $v \in \mathbf{R}^2$ is orthogonal to both $x$ and $y$, then $v$ is the zero vector. Here's my proof: Since $x$ and $y$ are ...
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Stability criterion for eigenvalues of an AR(2) process.

This is pretty much a question on linear algebra stemming from time series analysis. Essentially I want to find a stationarity criterion for an AR(2) process. It is easy to reduce this to the ...
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Solve $A_{1}KA_{2}-BK+KC=0$, for a $K \in R^{(1 \times n)}$, where $A_{1} \in R$ , $A_{2} \in R^{(n \times n)}$, $B \in R$, $C \in R^{(n \times n)}$.

Solve $A_{1}KA_{2}-BK+KC=0$, for a $1 \times n$ dimension row vector $K$, where $A_{1} \in R$ , $A_{2} \in R^{(n \times n)}$, $B \in R$, $C \in R^{(n \times n)}$ are known. How can I find the value ...
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37 views

Receiving different answers

Ok, so im following a tutorial on how to calculate a limit numerically and when the tutor plug'd in the number $(-1.1)$ into the equation $\frac{(t^6 -1)} {(t^3 + 1)}$ HE gets −2.331 as the ...
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1answer
34 views

$S_1 \subset S_2$. To show, $Span(S_1) \subset Span(S_2)$

Prove that if $S_{1} \subset S_{2}$, then $Span(S_{1}) \subset Span(S_{2})$ Approach: Suppose $S_{1} \subset S_{2}$ Let $x \in S_{1}$, then by definition of a subset, $x \in S_{2}$ All possible ...
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24 views

simultaneous diagonalize

$A = \begin{pmatrix} 18 & -9 \\ -9 & 9 \end{pmatrix}$ $B = \begin{pmatrix} 3 & 2 \\ 2 & -2 \end{pmatrix}$ Find a real invertible matrix such as $P^tAP = I_2$ and $P^tBP$ is diagonal ...
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How to find the inverse of the Haar (4) matrix? [on hold]

$$ H_4 =\begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & 1 & -1 & -1\\ 1 & -1 & 0 & 0\\ 0 & 0 & 1 & -1 \end{bmatrix} $$
3
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3answers
80 views

Is $\rho(A^2) = \rho(A)^2$?

How can I show that $\rho(A^2) = \rho(A)^2$? Is that even true? I´ve tested it with matlab for random matrices, and the equation was always true. I´m pretty sure that even $\rho(A^n) = \rho(A)^n$ ...