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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Does the iteration $e_i^\top x_{t+1} = \max_j e_i^{\top} (\alpha A^j x_{t} + b^j)$ converge?

Given a constant $0 < \alpha < 1$, a matrix $A \in R^{n \times n}$ and a vector $b \in \mathbb{R}^n$, it is well-known that a sufficient condition for the iteration $x_{t+1} = \alpha A x_t + b$ ...
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Finding inverse in this case?

Define a linear transformation $T\colon \mathbb{R}^3\to\mathbb{R}^3$, such that $T(x) = [x]_B$ ($B$-coordinate vector of $x$). $B = \{b_1, b_2, b_3\}$, which is a basis for $\mathbb{R}^3$. $b_1 = (...
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3answers
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Is it true that every element of $V \otimes W$ is a simple tensor $v \otimes w$?

I know that every vector in a tensor product $V \otimes W$ is a sum of simple tensors $v \otimes w$ with $v \in V$ and $w \in W$. In other words, any $u \in V \otimes W$ can be expressed in the form$$...
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34 views

Predicting a score for a test?

I don't know what I am doing wrong. Can anyone explain what goes in these matrices?
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379 views

Prove if $A$ is a square matrix and $AB=AC \Rightarrow B=C$, then $A$ is invertible.

Prove: if $A$ is a square matrix and $AB=AC$ implies $B=C$, then $A$ is invertible. First year linear algebra, haven't gotten to determinants yet so the proof can't use determinants or anything ...
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1answer
36 views

Zero Tensor Product

Suppose we have a space $|\psi_1\rangle \otimes |\psi_2\rangle \otimes |\psi_3\rangle$, and operators (matrices) A ⊗ B ⊗ C acting on this Hilbert space (like in quantum mechanics). I'm trying to ...
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76 views

Eigenspace of $T(A)=A^t$ on $\mathcal M_{n\times n}$

We have the map $T(A)=A^t$ as defined for $\mathcal M_{n\times n}$, which represents all $n\times n$ matrices. We know that an eigenvector of $T$ is a non-zero $n\times n$ matrix $A$ such that $A^t=\...
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1answer
73 views

Eigenspaces and eigenvalues of $T(y)=t\frac{dy}{dt}$

We have the map $T(y)=t\frac{dy}{dt}$ defined on $\mathbb R[t]_n$, which is the set of real-valued polynomials up to degree $n$, and I must find its eigenvalues and eigenspaces. In this case, we have ...
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I need help with creating linear equations from multiple points on a graph

How do you create a linear equation from multiple points on a graph I am working on a question where the points are $(-3,8),(2,5)$ and $(7,2)$ and I need to find out how to create a linear inequality ...
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1answer
59 views

$X = ABA^T$, $X$ is PSD and $B$ is Symmetric. Does $B$ have to be PSD to satisfy this equation?

Assume that I have given $X = ABA^T$. Also assume $X$ is PSD and $B$ is Symmetric with all diagonal elements equal to 1. Does $B$ also have to be PSD to satisfy this equation? Edit1: PSD means ...
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63 views

Seeing the plane as a four (or more) dimensional vector space on $\mathbb Q$

As I was trying to answer a question about the enumeration of circuits one can build with a set of miniature train track elements, I realized that all plane positions that could be reached had ...
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1answer
34 views

Altering diagonal entries of a square matrix to get simple eigenvalues

Let $A$ be an $n \times n$ matrix. Is there a diagonal matrix $D$ such that $A + D$ has $n$ distinct eigenvalues?
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77 views

Centralizer in O(2,R)

We have an ortogonal group of degree 2 over R. What is the order and structure of centralizer of subset which is any reflection from O(2,R)? As far as I know ortogonal group over a finite group is a ...
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If $\mathbf x\in \mathbb R ^n$ is a unit vector, why $\mathbf x^TN\mathbf x =1$ where $N$ is positive-definite and symmetrical?

Let $N$ be a $n\times n$ real symmetric and positive-definite matrix, and let $\mathbf x$ be a unit vector that satisfies $A\mathbf x = \lambda N\mathbf x$, where $A$ is a real symmetric matrix. Then ...
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1answer
57 views

Conditioning of Triangular Matrices:

Let $U \in \mathbb{R}^{N\times N}$ be upper triangular. $U$ is well conditioned if the magnitude of the diagonal elements is sufficiently large compared to that of the corresponding off-diagonal ...
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How to find Eigenspaces and Eignenvalues of a Map

I need to find the eigenspaces and eigenvalues of the map $T(y)=t\frac{dy}{dt}$ I know that this means that for eigenvalues $\lambda y=t\frac{dy}{dt}$, so then I integrate and get $\lambda=\frac{...
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1answer
29 views

Dimension of a vector subspace

Let $V=\mathbb{R}^{3}$ be a vector space. Let $U = \left\{(x, y, z)\in V \colon x+y+z=0\right\}$ and $W=\left\{(x,y,z)\in V\colon x-2y+z=0\right\}$ be subspaces of $V$. Find the dimension of the ...
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2answers
36 views

linearly dependent eigenvectors for a different eigenvalues

I can't seem to find a straightforward answer anywhere. Can eigenvectors of different eigenvalues be linearly dependent?
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127 views

Prove if A and B are n x n upper triangular matrices, so is AB

I'm trying to practice proofs for my linear algebra final and I've been stuck on this one for some time. I have $AB = [A\mathbf{b_1} \ A\mathbf{b_2} \ \dots \ A\mathbf{b_n}]$. I can show that $A\...
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1answer
62 views

How to compute the minimum possible sum?

Given two sets of having equal number of unique numbers we need to find the minimum possible sum. Where sum is the square of the difference of the number taken one at a time as a pair and each number ...
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3answers
71 views

Suppose A is a 5 x 5 matrix and suppose that det(A) = 0.

I have three questions... 1.) What could be said of the dimensions of the row space, column space, and the null space? I have that the dimeniosn of the row and column space would have to be less ...
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1answer
114 views

Let A be an n × m matrix with rank(A) < m and m ≤ n… How do I prove this?

Prove that it is always possible to write A = QR, where Q is an n × m matrix with orthonormal columns and R is upper-triangular?
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Geometric meaning of this inequality $|\det(A)|\leq ||a_1||\cdot||a_2||…||a_n||$

I was given this problem: Prove that $|\det(A)|\leq ||a_1||\cdot||a_2||...||a_n||$, where $A$ is an invertible matrix. I have managed to prove it by myself (using QR decomomposition), but I can't see ...
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1answer
78 views

Describe the conjugacy classes of UT(3,Zp)

How can all the conjugacy classes of UT(3,Zp) be described, if UT(3,Zp) is a unitriangular matrix group so its members look like $ \left( \begin{array}{ccc} 1 & a_{12} & a_{13} \\ 0 & 1 &...
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4answers
73 views

Let $V = \mathbb{R}^2$, show that $V$ is a vector space

I am new to the concepts of vector spaces (as far as I remember), and I have some difficulties in understanding how can I show, in practice, if a set is a vector space or not. I have an exercise in ...
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3answers
438 views

What is the condition for the linear system to be consistent

consider a linear system as follows $a_{1,1}\ x_1+a_{1,2}\ x_2+...+a_{1,n}\ x_n = b_1$ $a_{2,1}\ x_1+a_{2,2}\ x_2+...+a_{2,n}\ x_n = b_2$ ... $a_{m,1}\ x_1+a_{m,2}\ x_2+...+a_{m,n}\ x_n = b_m$ in ...
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2answers
73 views

How to find all eigenvalues of this monster (complex) matrix?

Let $$a=\cos(\frac25\pi)+i\sin(\frac25\pi)$$ and $$A=\begin{pmatrix}1&1&1&1&1\\1&a&a^2&a^3&a^4\\1&a^2&a^4&a^6&a^8\\1&a^3&a^6&a^9&a^{12}...
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How can you tell that a general solution to a DE is general?

At school, or in a first-year course on DEs, we learn (perhaps in less abstract language) that if you have a linear $n$th-order differential equation $$Ly = f$$ then the general solution is something ...
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1answer
43 views

Is that formula provable?

Is the following formula provable?: $0<a\Longrightarrow(a<b\Longleftrightarrow\frac{1}{b}<\frac{1}{a})$ where $a,b$ are real numbers
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1answer
74 views

Linear Algebra, Hoffman and Kunze's book. Chapter about linear functionals

Can anybody please help me to solve? Let $F$ be a field. We define $n$ linear functionals on $F^n$, for $n \geq 2 $, by: $f_k(x_1,\ldots,x_n) = \sum \limits_{j = 1}^n (k-j)x_j $. What is the ...
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Show that there are no values of a such that span{$u_1,u_2,u_3$} is a line in $\mathbb{R}^3$ that passes through the origin.

Let $u_1=(1,-1,a)$, $u_2=(a,0,1)$, $u_3=(1,1,a)$. (a) Show that there are no values of a such that span{$u_1,u_2,u_3$} is a line in $\mathbb{R}^3$ that passes through the origin. I figured out that $...
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$R$ be a commutative ring which is a vector space over some field $F$ , is the map $f(x)=rx , \forall x \in R$ $F$- linear for every $r \in R$?

Let $R$ be a commutative ring which is a vector space over some field $F$ ; for $r \in R$ consider the function $f:R \to R$ defined as $f(x)=rx , \forall x \in R$ , is this function a vector space ...
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680 views

1985 Putnam A1 Solution

I dont see what they mean by bijection of triples of subsets of $\{1, \ldots, 10\}$ and the $10\times3$ matrix with $0, 1$ entries? How is that created?
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Rotational matrices

I apologize ahead of time that math isn't my strong suit, I understand most the basic concepts but lots of gaps. So forgive me if i miss use a concept. So I am working in a 3d engine integrating a ...
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1answer
46 views

Show that composition of endomorphisms share some eigenvector

Let $\psi$ and $\varphi$ be endomorphisms such that $\psi$ $\circ$ $\varphi$ = $\varphi$ $\circ$ $\psi$ over some algebraically closed field. Show that they share an eigenvector. It seemed quite ...
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how I do I find the matrices of these linear transformattions

I asked this questions two days ago on Stackoverflow, and was advised to migrate it here. The problem is the way the bases are given is not in the usual way. It is rather confusing. Help will be ...
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2answers
45 views

solution of a system of equation

Let $A\in M_{m\times n}(\Bbb R)$ and let $b_0\in \mathbb R^m$. Suppose that the system of equations $Ax=b_0$ has a unique solution. Which of the following is true? $Ax=b$ has a solution for every $b ...
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1answer
56 views

Does a number matrix have its invariant factors??

I'm just confused by the following statement in my advanced algebra textbook: Frobenius Form: Let $A$ be an $n$th order square matrix over a number field $K$, whose invariant factors are: $$1,\...
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1answer
92 views

Perron-Frobenius theorem applied to continuous-time dynamical systems

This is a question about terminology rather than mathematics per se. I'm publishing a series of papers in which I make use of a fairly basic result that allows me to apply the Perron-Frobenius ...
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1answer
33 views

Finding basis for kernel and range

Define a linear transformation $T : P_3\to\mathbb{R}^4$ by $T(p(t)) =(p(0), p(0), p(0), p(0))$ I need to find a basis for the kernel of $T$ and the range of $T$. Can someone verify if the answers ...
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1answer
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Home depreciation real estate problem

"A property sells for $96,000$ USD . If it has appreciated $4$ percent per year straight line for the past five years, what did the owner pay for the property five years ago?" The answer listed in ...
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2answers
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Two vectors moving towards the same point - ensuring they both hit that point at the same time

I'm working on an algorithm which involves two vectors in 3D space. They're both moving towards a single point within their respective directions - I need to make sure that they both hit the same ...
4
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1answer
242 views

Factor $x^{14}+8x^{13}+3$

I need to factor this over the rationals, and there is a hint to use reduction mod3. The reduction is $x^{14}+2x^{13}=x^{13}(x+2)$, but I know it has no rational roots (they would have to be $\pm 3$ ...
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3answers
53 views

Finding eigenvalues using triangulars

I understand that if a matrix is a triangular, then the diagonal entries are eigenvalues of the matrix. If I row reduce any matrix to its triangular format, have I found all of the eingenvalues?
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Orthonormal basis of polynomials

I am trying to find an orthonormal basis of the vector space $P^{3}(t)$ with an inner product defined by $$\langle f, g \rangle = \int_0^1f(t)g(t)dt$$ by applying the gram schmit alogorotin to ${(...
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Does {(1, -1),(2, 1)} spans R2? please correct me

Can anyone please correct me? my problem is in the proof part below Q: Does {(1, -1),(2, 1)} spans R2? A: c1(1, -1) + c2(2, 1) = (x, y) c1 + (2)c2 = x -c1 + c2 = y ______________ c1 = x - (2)...
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45 views

Which of these two is an inner product space

I have a simple question on which am seeking help clarifying. Am looking at two inner products, one which my text says is an inner product and I find it to be not, and another my text says it is but I ...
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1answer
47 views

Factorisation of Sigma Series question

How would I factorize the following? For people who can't read my hand writing it's 3/2n(n+1)(2n+1)-2n(n+1)+n, the equation comes from a sigma series question I have worked up to but I am really ...
4
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4answers
601 views

Given that it appears impossible to make the set out of linear combinations of its elements, why is it still dependent?

The answer key says the following set of functions is linearly dependent: $\{5, \cos^2x, \sin^2x\}$. Without calculating the Wronskian, I would've guessed it was independent because there's ...