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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How many ways are there to prove Cayley-Hamilton Theorem?

I see many proofs for Cayley-Hamilton Theorem in textbooks and net, so I want to know how many proofs are there for this important and applicable theorem.
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4 views

Find a basis for the image of the given linear transformation

Find a basis for the image of the following linear transformation $$ t:\mathbb{R}^3 \rightarrow \mathbb{R}^3$$ and $$ t(x,y,z)=(x+3y-2z,x-y+z,3x+y)$$ I have found the following basis: $$ \left\{ ...
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0answers
33 views

Finding complex eigenvectors

Can anyone help me to point out what I am doing wrong? I need to find a change of bases matrix for the complex eigenvalue (so I can find closed formula). I was successful in finding eigenvalues, ...
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1answer
31 views

Is the following set of vectors orthonormal?

$\left( \begin{bmatrix}\frac{2}{\sqrt{10}}\\\frac{2}{\sqrt{10}}\\1\\1\end{bmatrix}, \begin{bmatrix}\frac{1}{\sqrt{3}}\\0\\-\frac{1}{\sqrt{3}}\\-\frac{1}{\sqrt{3}}\end{bmatrix}, ...
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0answers
23 views

What is an algebraic eigenvalue?

Is an algebraic eigenvalue just an eigenvalue that is an algebraic number? I have seen this term in multiple places but am unsure (not the same thing as an eigenvalue's algebraic multiplicity).
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2answers
35 views

Linear algebra endomorphism

$$ f(A) = A \begin{pmatrix} \alpha & 1 \\ 1 & 1 \end{pmatrix} - \begin{pmatrix} \alpha & 1 \\ 1 & 1 \end{pmatrix} A $$ How do you calculate the characteristic polynomial, the ...
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2answers
504 views

In each part, determine whether the three vectors lie in a plane in $\mathbb{R}^3$

I could not find the part that told me how to find out this question at all. I do know that three vectors in $\mathbb{R}^3$ are linearly independent if and only if they do not lie in the same plane ...
5
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0answers
40 views

Algebra defined by $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$

Let $\cal A$ be the (noncommutative) unitary $\mathbb Z$-algebra defined by three generators $a,b,c$ and four relations $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$. Is it true that $ab\neq 0$ in $A$ ? This ...
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2answers
16 views

Prove that $\frac{(\sum_{i=1}^m \|s_i - p\|_1)}{\sqrt2} \leq \sum_{i=1}^m \|s_i - p\|_2 \leq \sum_{i=1}^m \|s_i - p\|_1$

Prove that $$ \frac{1}{\sqrt2} \sum_{i=1}^m \|s_i - P\|_1 \leq \sum_{i=1}^m \|s_i - P\|_2 \leq \sum_{i=1}^m \|s_i - P\|_1 $$ Where $m$ is a number of points in 2d plane, e.g $s_i = ...
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1answer
15 views

Reducible matrices and strongly connected graphs

The incidence matrix below is primitive (the graph is strongly connected) thus it is irreducible. However, this matrix looks reducible (using $P = I_n$) ?! What am I missing? [A is called reducible ...
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26 views

Find a matrix p diagonalizes A and determine $p^{-1} A p$

Find a matrix $p$ diagonalizes A 3×3 matrix and determine $ p^{-1} A p $ $$ A = \begin{bmatrix} 2 & 0& -2 \\ 0& 3& 0 \\ 0& 0& 3 ...
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2answers
35 views

The graph of the linear function $t$ has intercepts at $(x,0)$ and $(0,y)$. If $x \neq y$ and $x+y=0$, is the slope negative or positive?

“The graph of the linear function $t$ has intercepts at $(x,0)$ and $(0,y)$ in the $xy$-plane. If $x\neq y$ and $x + y = 0$, which of the following is true about the slope of the graph of $t$?” I was ...
4
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2answers
190 views

How to list all possible dimension of $\ker{T},\ker{T^2},…,\ker{T^{k-1}}$ and the corresponding canonical forms?

Let $V$ be $5$-dimension vectorspace, and $T:\ V\rightarrow V$ a nilpotent linear transofrmation of order (index) $k$ where $1\le k\le 5$. How to list all possible dimension of ...
2
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2answers
409 views

Can I use determinants to show that two vector sets span the same subspace?

I have two sets of vectors, like these: $v_1 = (1, 6, 4)$ $v_2 = (2, 4, -1)$ $v_3 = (-1, 2, 5)$ in set $V$ $w_1 = (1, -2, 5)$ $w_2 = (0, 8, 9)$ in set $W$ I need to show that $V$ and $W$ span the ...
2
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1answer
29 views

Prove that $J_n(0)$ and $(J_n(0))^t$ are similiar

Prove that $J_{n}(0)$ and $(J_{n}(0))^t$ are similar ($J_n(0)$ is a $n \times n$ Jordanian block which belongs to the eigenvalue $0$). Use your answer and Jordanian form to prove that every matrix $A ...
1
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1answer
2k views

Least Squares Plane using Matricies

For a Least Squares solution to a 2D set of coordinates, the formula is: $X^T\,X\,\vec b = X^Ty$ (where $X^T$ denotes $X$ transpose) (for: $y = B_0 + B_1x + B_2x^2$) where: ...
3
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1answer
34 views

The matrix square root is not differentiable on the boundary of the manifold of positive semi-definite matrices?

$\newcommand{\psym}{\operatorname{P}_{\ge 0}}$ $\newcommand{\Sig }{\Sigma}$ Let $\psym$ denote the subset of symmetric positive semi-definite matrices. Let $S:\psym \setminus \{0\} \to \psym ...
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0answers
31 views

Geometric explanation of a methodology in the article about Image Denoising

In article Ghimpeteanu G., et al. - A Decomposition Framework for Image Denoising Algorithms, I found as below: Let $\displaystyle I : \Omega \subset R^2\mapsto R$ be a gray-level image, and $(x, ...
3
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2answers
39 views

Reduction of basis and dimensions

If $S$ is a $5-$dimensional subspace of $\mathbb{R}^6$ .Is it true that every basis of $\mathbb{R}^6$ can be reduced to a basis of $S$ by removing one vector ?
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2answers
48 views

Prove positive definite of a function

For $A,X,Q \in \mathbb{R}^{n \times n}$, define $h(X) = A X A^T + Q$ and $ h^j(X)=\underbrace{{h(h(}...h}_{j\text{ times}}(X)))$. If $X,Q$ are positive definite, $A\neq 0$ and for a certain integer ...
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0answers
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How to solve exponential matrix factorization with constrain: $UV^T>0$

recently I would like to optimize the following loss function: $$L=\sum_{ij}W_{ij}(X_{ij}-exp(-\sum_{l} U_{il}V_{jl}))^2$$ $$s.t. \sum_lU_{il}V_{jl} > 0$$ Where $W \in \mathbb{R}^{m \times n}, X ...
0
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1answer
96 views

For these subsets $S$, are they subspace for the indicated vector space $V$

Q1. $V =P_5(R)$ and $S=\{p(x)\mid p(15)=0\}$. I think it is a subspace, but not 100% sure. I tried let $p_1(x)=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5$, such that $p_1(15)=0$ ...
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0answers
29 views

Concerning Linear Algebra on Two Finite Fields…

I am interested in performing linear algebra modulo 6, which I believe can be split into linear algebra modulo 2 and linear algebra modulo 3. I'm hoping that someone can provide some insight into ...
2
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2answers
33 views

Prove that $ABA^T$ is symmetric when $A$ and $B$ are symmetric matrices

I have been learning about matrix symmetry and came up with a question that I can't seem to prove. The idea is that the product of $ABA^T$ is a symmetric matrix. What I mainly have to go off of is ...
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0answers
18 views

Prove that this transformation of a stochastic matrix (or markov chain) is still a stochastic matrix (or markov chain)

assume that I have an $N \times N$ stochastic matrix $W$, where $\sum_j w_{ij} = 1$ and $w_{ij}$ is a generic element on row $i$ column $j$ of the matrix $W$. Moreover I have the following two $N ...
2
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1answer
45 views

What we can tell about complex matrices? Ideas for a school work

Background: I have some background in abstract and linear algebra. In my undergrad complex calculus class, I have to write a $5$ page paper about "complex matrices". I don't know exactly what the ...
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28 views

What is the dimension of $\ker f =\{(x^3-x)Q(x):Q \in\mathbb{R}_{n-3}[x]\}$?

I have $$\ker f =\{(x^3-x)Q(x):Q \in\mathbb{R}_{n-3}[x]\}.$$ Here $f$ is the following endomorphism $$f(P) = (x^2-x+1)P(-1)+(x^3-x)P(0)+(x^3+x^2+1)P(1),$$ where $P\in\mathbb{R}_{n}[x]$. My ...
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1answer
19 views

Can you extract the horizontal component of the change of two quaternions?

I receive orientation data as quaternions, and I'm interested in finding the ground-planed component of the change in angle. I know that the arccosine of the dot product of two quaternions gives me ...
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1answer
33 views

Norm of $xy$ equals norm of $yx$

Given is that $\|y\| = \|x\|$, Is it true that $x^Ty = \|xy\|$ and $y^Tx = \|xy\|$ If the things above are true, is this also true $\|xy\| = \|yx\|$? And why or why not ?
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1answer
14 views

Use of “iff” in this statement about subspaces?

"$H$ is a subspace of $V$ iff for all vectors $x, y \in H$, it follows that $x+y$ is also in $H$" This is a true or false question. Now of course a necessary property of a subspace is that it is ...
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0answers
21 views

What are the basis vectors of the cone of positive semi definite matrices?

I was wondering if we could find a set of basis vectors that span the cone of positive semidefinite matrices? I know this question is hard, but I would really appreciate if even someone can share ...
0
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1answer
12 views

Weird transposing after dot product and transformation

I'm reading a paragraph in a book where a plane equation ($N\cdot Q + D = 0$, N being the normal and D the distance from the origin, Q any point which belongs to the plane) is transformed by a matrix ...
1
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1answer
60 views

Why does rotation by a quaternion require multiplying two times?

Given a vector $p$, to rotate it by a quaternion $q$, we use the formula: $$p' = q p \hat{q}$$ where $\hat{q}$ is the conjugate of $q$. But if we use rotational matrices, then it's just $$p' = ...
0
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1answer
29 views

isomorphism from one vector space to another one

This is from my textbook I don't quite understand what isomorphism means. Greek word "isomorphism" means same structure, but how can we say $P_3$ has the same structure as $R^4$?
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1answer
71 views

Without using row operations, and by inspection, Find one nontrivial solution to Ax=0

Link to the image of the problem Hi, I am stuck on this problem. Can anyone explain to me how I can come to a conclusion.
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0answers
14 views

Jacobi SVD algorithm implementation

Is this implementation of Jacobi SVD algorithm according to the standard algorithm? Please verify. I have seen pseudo code of Jacobi algorithm like here which appears quite different from how it is ...
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3answers
23 views

Reduced row echelon form of matrix with trigonometric expressions

I'm trying to solve for the eigenvalues of and the eigenvectors of a rotation matrix (about the z-axis): $$A = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & ...
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23 views

map extension in linear transformations

Let $V$ be a finite dimensional vector space over a field $F$ and $W$ be a subspace of $V$ If $T:W \to U$ is a linear map for some vector space $U$ over $F$ then there exists a linear map $S:V ...
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0answers
19 views

Derivative of L2 norm

I am reading a paper about image processing and I have a question. In the paper we have equations like below. $X_{C1} = 0.596X_R - 0.274X_G - 0.322X_B$. $X_{C2} = 0.211X_R - 0.523X_G + 0.312X_B$ ...
5
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1answer
263 views

Problem with sum of projections

Let $X$ be a real linear space, $(P_i)_{i=1}^n$ -a finite sequence of linear mappings $P_i :X\rightarrow X$ such that $P_i^2=P_i$ for $i=1,...,n$, $(P_1+...+P_n)^2=P_1+...+P_n$. I wish to show ...
2
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1answer
31 views

Linear algebra over $\mathbb{Z}$

Suppose I have $v_1,\ldots,v_n$ vectors in $\mathbb{Z}^n$. Let $M$ be the matrix whose columns are $v_1,\ldots,v_n$. I would like to know if, as it happens with a vector space over a field, $M$ is ...
0
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1answer
51 views

If $\,ax+b=cx+d,\,$ then is $\,a=c\,$ and $\,b=d$?

I am a high school student my maths teacher said that if $\,ax+b=cx+d,\,$ then is $\,a=c\,$ and $\,b=d.\,$ Can someone give me a prove of this?
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0answers
28 views

null space from SVD

It is said that a matrix's null space can be derived from QR or SVD. I tried an example: $$A= \begin{bmatrix} 1&3\\ 1&2\\ 1&-1\\ 2&1\\ \end{bmatrix} $$ I'm convinced that QR (more ...
3
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1answer
26 views

What can the operator $T$ be?

I am trying to solve 3c from this released exam: Determine all operators $T \in \mathcal{L}(V)$ such that $T^3 = T$ and $T^* = -T$. What can $T$ be? From part b I have deduced that the ...
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0answers
17 views

Linear independent in random variable and observations

I am confused with some fundamental concepts. Here for $n$ random variable $X_1,\cdots,X_n$, i.i.d and follow standard normal distribution, the probability that there exists a set of constant ...
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1answer
37 views

Continuously differentiable functions

Let $f, g,$ be $ C^2$ functions $\mathbb{R} \rightarrow \mathbb{R}$, $ F: \mathbb{R}^2 \rightarrow \mathbb{R}, F(x,y) = f(x+g(y))$ Check that $(D_1F)(D_{12}F)=(D_2F)(D_{11}F)$ I know how to ...
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2answers
21 views

Hermitian and positive eigenvalues implies positive-definite

On Wikipedia it says that for Hermitian matrices, positive eigenvalues if and only if positive-definite. How do you prove the forward direction?
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1answer
72 views

Does this form of matrix have a name?

I'm looking for the name of this kind of $n$-by-$n$ matrix: $$\left(\begin{array}{cccc} -s_1 & b_{12} & b_{13} & b_{14} \\ b_{21} & -s_2 & b_{23} & b_{24} \\ b_{31} & ...
0
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1answer
14 views

maximum norm in an inner product space

the question: Let $V$ be the vector space of all continuous functions over the interval $[0,1]$. Does an inner product space over $V$ that define this norm: $|f|_{\infty}=max_{0\le x\le1}|f(x)|$ ...
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1answer
7 views

Does LU factorization needs pivoting?

Today I have a numerical methods exam,and of course i tried some exercices, but today I heard something that messed my mind, I always do LU fact. Like this : I take Lower triangular matrix, and then ...