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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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
34 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
14 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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1answer
27 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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10 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
11 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
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
9 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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24 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
24 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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21 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
35 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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27 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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20 views

Under What Intervals Is A Matrix Positive Definite, Positive Semi-Definite, Indefinite, Negative Definite and Negative Semi-Definite? [on hold]

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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7 views

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
45 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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5 views

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
41 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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3answers
27 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 ...
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1answer
24 views

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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53 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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2answers
24 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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4 views

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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16 views

Solve $KA-BK=0$, for a $1 \times n$ dimension row vector $K$, where $A$ is known $n \times n$ matrix and $b$ is known scalar

The above equation with mentioned dimensions is to be solved. How can I find the value (or approximate value) of row vector $K$. Please help.
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36 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
31 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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21 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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26 views

How to find the inverse of the Haar (4) matrix? [on hold]

$$ H_4 =\begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & 1 & -1 & -1\\ 1 & -1 & 0 & 0\\ 0 & 0 & 1 & -1 \end{bmatrix} $$
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3answers
78 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 $\rho(A^n) = \rho(A)^n$ ...
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1answer
19 views

Picture of vector in $R^3$ and vector in $R^2$ reflected across plane [on hold]

I have trouble imagining what reflecting a vector in $R^2$ and a vector in $R^3$ across x-y plane and y-z plane look like. Would you please draw me a picture?
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1answer
32 views

Does $A-\lambda I$ have rank smaller than $A$?

Consider $\lambda$ as one eigenvalue of $A$, can we say that $A-\lambda I$ must have rank smaller than $A$? Or equivalently, $A-\lambda I$ spans a space which is a subset of $A$?
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1answer
15 views

Linear independence and Wronskian - Proof or Counterexample

If $y_1(x) , y_2(x) ,\ldots,y_n(x)$ are linearly independent in $C[b,c]$ then they are Linearly Independent in $C[a,d]$, where $a<b<c<d.$ So I know if the Wronskian isn't zero for at least ...
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5 views

Gradient Descent: L2 Norm Regularization

So I've worked out Stochastic Gradient Descent to be the following formula approximately for Logistic Regression to be: $ w_{t+1} = w_t - \eta((\sigma({w_t}^Tx_i) - y_t)x_t) $ $p(\mathbf{y} = 1 | ...
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22 views

Find only the real eigenvalues of a matrix.

If a matrix has many (thousands) complex and few (dozen) real eigenvalues is there a fast method for estimating only the real eigenvalues ?
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1answer
25 views

Find the vector form of the equation of the line in $\mathbb{R}^2$ that passes through $P = (2, -1)$

Find the vector form of the equation of the line in $\mathbb{R}^2$ that passes through $P = (2, - 1 )$ and is parallel to the line with general equation $2x - 3y = 1$. Following the format of $x = p ...
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30 views

Verify rotation relation between two matrices

Suppose we have two matrices how do we verify that one of them is related to the other by a rotation, $$AU = B$$ where $UU^T=I$. One way is to form $AA^T$, and $BB^T$ and see if they are equal. How ...
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79 views

Determinant of this matrix? [on hold]

How can I find the determinant of this matrix?
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0answers
23 views

The sum of two subspaces

Let $V_{1}$ and $V_{2}$ be two subspaces of V. Define the sum of $V_{1}$ and $V_{2}$ to be the subset of V $V_{1}+V_{2}=${$\overrightarrow v_{1} + \overrightarrow v_{2}:\overrightarrow v_{1} \in ...
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1answer
10 views

Magnitude of orthogonal projection

I have a basic linear algebra question. Suppose that $ u \in \mathbb{R}^n $, and $ P(u, V) $ is the orthogonal projection of $ u $ onto a linear subspace $ V $. I would like to prove that $$ ...
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28 views

System with parameters [on hold]

How can I solve the following system where $\lambda$ and $\mu$ are parameters and what is the answer?
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27 views

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 ...
4
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1answer
44 views

Number of invertible matrices modulo 26

The number of invertible matrices modulo $26$ can be computed by the Chinese Remainder Theorem. i.e. a matrix is invertible modulo 26 if it is invertible modulo $13$ and modulo $2$ which are given ...
4
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1answer
62 views

Finding the Determinant of a particular Matrix

I've come across the question of finding the determinant of the $(n\times n)$ matrix, given by $$A:= \begin{pmatrix} x & 1 & 1 & \dots & 1 \\ 1 & x & 1 & \dots & 1 \\ ...
3
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2answers
34 views

Is $\text{Rank }(T) = \text{Dim}(V)$ all the time?

Thm Let $V$ and $W$ be Vector spaces and let $T:V \to W$ be linear If $\beta = \{ v_1,\dots ,v_n \}$ is a basis for $V$ then $$ R(T)=\text{span}(T(\beta))=\text{span}(\{ T(v_1),\dots,T(v_n) \} ...
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0answers
43 views

Solution of $x'=Ax$ is not what it is supposed to be

Consider a system of ODEs $x'=\begin{bmatrix} 0 &1 \\ -1&0 \end{bmatrix}x$. Wolfram Alpha says that the solution is $x(t) = \begin{bmatrix} \cos t &\sin t \\ -\sin t& \cos t ...
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3answers
25 views

Understanding Defiinition of Vector Space

Let $F$ be a field. A vector space over $F$ is a set $V$ together with $+$,$\cdot$ satisfiyng: $$+: V \times V \rightarrow V$$ $$\cdot: F \times V \rightarrow V$$ with usual properties. My ...
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33 views

Minimizing the error by finding optimum step-size

I need to recheck a proof for minimizing the error by finding optimum step-size. I re-checked the proof many times but still can't find a mistake although the number I am getting in Matlab is not ...
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11 views

Perturbation theory for a symetric rank-one update

I know perturbation theory of the eigenspectrum/singular value decompostion of a symetric matrix $A$ under a symetric perturbation $E$, that besides being symetric has no other structure. Is there ...
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37 views

$ax=0$ if and only if $a=0$ or $x=0$ [duplicate]

Prove that $ax=0$ $\Leftrightarrow$ $a=0$ $\lor$ $x=0$, where $a$ is a scalar from a field and $x$ is an element of the vector space on this field. I would like a hint or maybe a solution to prove ...