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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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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26 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
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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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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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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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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4answers
5k views

Raising a square matrix to a negative half power

I want to implement the following formula (taken from Kaiser, 1970) in R where $R$ is square matrix of correlations: $$S = (\textrm{diag } R^{-1})^{-1/2}$$ I understand the diagonal and inverse ...
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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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39 views

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 $A,B$ $n\times n$ ...
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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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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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400 views

determinant recursive formula of a specific matrix

For a field $K, n \in \mathbb{N}_{>0}$ and $\lambda \in K$ let $A_{n, \lambda} \in \textrm{Mat} (n,K) $ be the following matrix with entries $\lambda$ on the diagonal, $-1$ on both minor diagonals ...
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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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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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2answers
792 views

The trace-determinant plane, classification of equilibria of differential equations

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form ...
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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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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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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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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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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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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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495 views

How to solve this percentage question without equations?

Question - The total population of a village is 5000. The number of males and females increases by 10% and 15% respectively and consequently the population of the village becomes 5600. What was the ...
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1answer
54 views

How to show that the axiom for vector space hold for the following operation?

So the operation is sum defined by $f+'g=f\circ g$ (composite of functions) and usual scalar multiplication. First, for $(x+y)+z=x+(y+z)$ property, $(f+'g)+'h=(f \circ g)\circ h\ne f \circ (g\circ ...
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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
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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1answer
43 views

Finding a vector space over $F$ of dimension $m$ and $n$

The question is below: Let $V$ and $W$ be vector spaces over $F$ of dimensions $m$ and $n$, respectively. Find a basis for $L(V,W)$. This is what I have: Let$ (v_1, v_2, \ldots, v_m)$ be a basis of ...
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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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0answers
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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1answer
30 views

How many subsets of unknowns whose sum can be determined by the underdetermined system $Ax=b$ with $A \in \{0,1\}^{m \times n}$

Consider a underdetermined system $Ax=b$ with $A \in \{0,1\}^{m \times n}$ (i.e. being a binary matrix), $x \in \mathbb{R}^n$ and $b \in \mathbb{R}^m$. I want find a set $S$, $e \in S$ if and only if ...
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1answer
43 views

Transformation matrix from principal angles and vectors

If I got it right, given two planes in $N$-dimensional space ($N\gg2$), their 2 principal angles ($\theta_1$, $\theta_2$) and 4 vectors ($\vec{a}_1$, $\vec{a}_2$, $\vec{b}_1$, $\vec{b}_2$) can be ...
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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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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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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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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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1answer
47 views

A generalized eigenvalue problem

The generalized eigenvalue problem likes this: $\begin{pmatrix} 0 & C_{12}\\ C_{21} & 0 \end{pmatrix} \begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}=\rho\begin{pmatrix} C_{11} & 0\\ 0 ...
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1answer
1k views

Equivalent systems of Linear equation

I've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...
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21answers
10k views

Can an infinite sum of irrational numbers be rational?

Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational. Is it possible for $S$ to be rational, considering the additional restriction that non of the $a_k$'s is a linear ...
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104 views

Let $S = \{n\in\mathbb{N}\mid 133 \text{ divides } 3^n + 1\}$. Find three elements of S.

Question: Let $S = \{n\in\mathbb{N}\mid 133 \;\text{divides} \; 3^n + 1\}$ $a)$ Find three different elements of $S$. $b)$ Prove that $S$ is an infinite set. My intuition is find the prime factors of ...
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How to find the inverse of the Haar (4) matrix? [on hold]

$$ H_4 =\begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & 1 & -1 & -1\\ 1 & -1 & 0 & 0\\ 0 & 0 & 1 & -1 \end{bmatrix} $$
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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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2answers
1k views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
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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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2answers
1k views

$A$ is skew hermitian, prove $(I-A)^{-1} (I+A)$ is unitary

Given $A$ is a skew-hermitian, (i.e $A^H=−A$), the Cayley transform of $A$ is defined as: $W=(I-A)^{-1} (I+A)$. How can be proved that $W$ is unitary (i.e. $W^H W = W W^H = I$)?
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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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4answers
177 views

Let A be a square matrix such that $A^3 = 2I$

Let $A$ be a square matrix such that $A^3 = 2I$ i) Prove that $A - I$ is invertible and find its inverse ii) Prove that $A + 2I$ is invertible and find its inverse iii) Using (i) and (ii) or ...
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1answer
445 views

The annihilator of an intersection

I know this question has been arlready asked, but as my reputation is too low I'm not allowed to post a comment, sorry for this second post. I'm asked to prove : $(W_1+W_2)^0=W_1^0\cap W_2^0$. ...