Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Bases and Matricies

Assume $ dim V = 2 $ and $\{e_1, e_2\}$ is a basis of $V$. Suppose that $$ M(T,\{e_1,e_2\}) = \left (\begin{array}{cc} 0 & 0 \\ 1 & 1 \end{array} \right) $$ Find a basis $\{v_1,v_2\}$ of $V$ ...
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Tensor product of two linear map and its matrix representation

Suppose $T_1: \mathbb{R}^n\to\mathbb{R}^n$ be any linear map and wrt a basis $\{e_1,\dots,e_n\}$ the matrix of $T_1$ is $M$, aand $T_2:\mathbb{R}^m\to\mathbb{R}^m$ be another linear map whose matrix ...
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35 views

Prove that $\det(A) > 0$

Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a real $n \times n$ matrix such that : $A^{3} = A + \mathrm{I}_{n}$. Prove that $\det(A) > 0$. Here is what I tried : $X^{3}-X-1$ is a null polynomial ...
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Problem on Finding the rank from a Matrix which has a variable

$$ A = \begin{bmatrix} 1 & -1 & -2 & -3 \\ -2 & 1 & 7 & 2 \\ -3 & 3 & 6 & \alpha \\ 7 & -6 & -17 & -17 \end{bmatrix} $$ Find the rank when ...
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36 views

Is there any easy way to see that elementary matrices commute in $\text {Mat}_{n \times n} (\mathbb F)$?

Is there any easy way to see that elementary matrices commute in $\text {Mat}_{n \times n} (\mathbb F)$ ? I've been trying to sketch a proof by induction, but it seems more complicated that it should ...
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ODE -parabolic cylinder functions

How do we solve $\frac{d^2f}{dz^2} + \left(Az^2+Bz+C\right)f=0 \tag 1$ where $f(z),A,B,C$ are matrices of order $3 \times 3$.
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1answer
8 views

Connection between linear independence, non-/trivial and x solutions

I am having a hard time remembering which goes hand in hand with what. The math questions I get always include words like trivial etc. 1 solution no solution infinite amount of solutions And then ...
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10 views

What is special about a transformation if the matrix of that transformation is symmetric?

If the matrix of a linear transformation T$\colon \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$ with respect to some basis is symmetric, what does it say about the transformation? Is there a way to ...
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Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points (X and Y), giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix ...
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1answer
26 views

Rank of a matrix and dimension of the image

I'm teaching linear algebra to first year students, and I was recently asked why is the rank of a matrix, representing a linear application in a given basis, equal to the dimension of the image space ...
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26 views

Sum Matrix example

I am reading in my Linear Algebra book and nowhere it states what exactly a "sum matrix" is. This is the closest part of the text that describes. The part I have painted in yellow is what I am ...
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38 views

Find dimension and basis of the set of all points in $R^5$ whose coordinates satisfy the relation $x_1+x_2+x_3+x_4=0$

Doesn't a basis in $R^5$ require 5 finite vectors to be a basis? I'm really confused, maybe theres something I am missing on how $R^5$ can have 4 coordinates that can add up to zero? Does that imply ...
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21 views

Possible dimensions of $A_{n\times n}$

What are the dimensions of $A_{n\times n}$ such that $A\vec{c} = 0$ for $\vec{c} \in \mathbb{R^n}$? I have that the dimensions possible are $n^2$ and $n^2-n$ since we can have $A = -\vec{c}$ on the ...
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Linear dependence of set of linear combinations of linearly independent vectors

I came across this problem: Given the set of linearly independent vectors $\{u,v,w\} \subseteq \mathbb{R}^n$, determine whether or not the following set of vectors is linearly dependent: $\{u-v-w, ...
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1answer
15 views

Rank of a linear transformation T

Let, n be a positive integer & let M_n(R) be the space of all n*n real matrices. If T:M_n(R)-->M_n(R) is a linear transformation such that T(A)=0, whenever A belongs to M_n(R) is symmetric or ...
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65 views

Find the coordinates of the expression $(\cos x + \sin x)^3$ in the basis {$1, \sin x, \cos x, \sin 2x, \cos 2x, \sin3x, \cos 3x$}

I'm quite stumped here. I expanded out $(\cos x + \sin x)^3$ to $\sin^3x + \cos^3x + 3\sin x\cos^2x + 3\sin^2x\cos x$ And I've tried the trig identities for $\sin 3x = 3\sin x - 4\sin^3x$ I can't ...
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31 views

Bases of Matrix

Suppose that $\{e_1, e_2\}$ is a basis of $V$, and dimension of $V$ is 2. Assume that $$ M(T,\{e_1,e_2\}) = \left [\begin{array}{cc} 0 & 0 \\ 1 & 1 \end{array} \right] $$ Find a basis ...
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26 views

{$e_1, e_2, e_3, e_4$} be a standard basis in $R^4$. Does there exist a set of vectors $x_1, x_2, x_3$ such that

sets {$x_1, e_2, e_3, e_4$}, {$e_1, x_2, e_3, e_4$} and {$e_1, e_2, x_3, e_4$} are basis but {$x_1, x_2, x_3, e_4$} is not a basis? I think the answer is no, but what if $x_4$ was a linear ...
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2answers
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Dimension of matrices with entries $a_{ij} = a_{rs}$ with $i+j = r+s$.

Let $n$ be a positive integer and $H_n$ be the space of all $n \times n$ matrices $A = (a_{ij})$ with entries in $\Bbb{R}$ satisfying $a_{ij} = a_{rs}$ whenever $i+j = r+s \; (i,j,r,s = 1, 2, \ldots, ...
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Converting from X,Y,Z offset representing a rotation to matrices

I've been working on figuring out 3-dimensional rotations for graphics and I've reached a brick wall of understanding that I can't power through. Right now I have a function which calculates the ...
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3answers
84 views

Prove that $A^{10}$ is equal to linear combination of $A^k, k = 1,…,9$ and identity matrix.

Let $A=\begin{bmatrix}2&0&1\\0&1&0\\1&0&1\end{bmatrix}$. I did this with brute force and it was messy, is there a more theoretical way?
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1answer
16 views

Connection between adjoint of a matrix and adjoint of an operator

Let $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ with $$T(x,y) = \left[ \begin{array}{ccc} 1x+2y \\ 3x+4y \end{array} \right] $$ The matrix representation of $T$ is $$ A= \left[ \begin{array}{ccc} 1 ...
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2answers
22 views

Let $B={b_1, b_2, … , b_n}$ be a basis for a vector space V. Prove that each v ∈ V can be expressed as a linear combination of $b_i$

such that $v = \alpha_1 b_1 + \alpha_2 b_2 + ... + \alpha_n b_n $ in only one way that the coordinates of $\alpha_i$ are unique. Am I right to assume that this question is asking to prove that B ...
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1answer
40 views

Is there an intuitive explaination of the matrix $A^T A$

Is there an intuitive explaination of the matrix $A^T A$? I have seen this in many field and it is also a matrix with a lot of good properties. Is there some intuitive explaination of it or a name for ...
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41 views

Let {$a_1, a_2, a_3$} be some basis in $R^3$. Does the set of vectors {$b_1, b_2, b_3$} form a basis if

a) $b_1$ = $a_1$ x $a_2$, $b_2$ = $a_2$ x $a_3$, $b_3$ = $a_3$ x $a_1$? (cross product) b) $b_1 = a_1$ x $(a_2$ x $a_3), b_2 = a_2 $x $(a_3$ x $a_1), b_3 = a_3$ x $(a_1$ x$ a_2)$? I got as far as ...
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matrix representation of the basis transform by Gram-schmidt orthogonalization

Gram-schmidt orthogonalization transforms a basis to an orthogonal basis. This transform of bases is linear. How can we write down the matrix representation of the transform then? Thanks.
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1answer
34 views

The residue of $x^{15}-1$ divided by $x^2-1$

The residue of $x^{15}-1$ divided by $x^2-1$ I want to use the long division, but I don't know how to start
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1answer
18 views

Proof of upper triangular matrices

I am supposing that $A=(a_{ij})$ and $B=(b_{ij})$ are two $n\times n $ upper triangular square matrices. $\lambda \in \mathbb{R}$. So $a_{ij}=0$ whenever $i>j$. I am trying to prove that these are ...
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Question in Regards to Basis, Polynomials, and coordinates

So I'm having some serious problems understanding this. Any help would be appreciated. Consider the polynomial f(x) = x^5 − 5x^4 . (a) Find coordinates of f' , f'' , f''' in the basis {1, x, x^2 , ...
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1answer
24 views

finding the derivative of g' and h'

So I know i'm not too terribly far off of the wrong answer but i'm not sure where I went wrong so I was just looking for a little help here. and sorry ahead of time but I don't know how to use the ...
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7 views

What is the change of basis in 2D?

I know how to apply a change of basis in 1D, but I was wondering: If I want to apply a change of basis to a nxn matrix, is it enough to apply the change of basis to every column of the matrix or is ...
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44 views

Dumb question about characteristic polynomial of a matrix

Can you row reduce the matrix before computing $\det(\lambda I-A)$? Will this still give an equivalent characteristic polynomial?
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22 views

Calculate angle betwen two lines

I have been trying to find the best solution to this problem, but my math is pretty bad. What I want to do is calculate the "Angle" in radians, I have all the 3 co-ordinates and all the 3 lengths ...
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21 views

Basis of Trigonmetric Polynomials Help

Write the following trigonometric polynomials in terms of the basis functions: $\cos^2(x)$ $\cos^2(x) \sin^3(x)$ Is there a certain way to solve these types of problems because I'm very unsure on ...
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1answer
21 views

Do the given vectors span $\mathbb{R}^3$?

Do the following vectors span $\mathbb{R}^3$: $$v_1 = (2, -1,3)$$ $$v_2 = (4, 1, 2)$$ $$v_3 = (8, -1, 8)$$ I use Gaussian Elimination to bring the matrix to an echelon form, with a pivot of "1" in ...
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2answers
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Find the coordinates of a vector (7,14,-1,2) in the basis (1,2,-1,-2), (2,3,0,-1), (1,2,1,4) and (1,3,-1,0) [on hold]

Not really sure what the basis (1,2,-1,-2), (2,3,0,-1), (1,2,1,4) and (1,3,-1,0)
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$(2,1+\sqrt{-5}), (1-\sqrt{-5},2)$ generate the $\mathbb Z[\sqrt{-5}]$-module $\langle 2,1+\sqrt{-5} \rangle \times \langle 2,1+\sqrt{-5} \rangle$

Ok, boring question here (I guess, at least). Let $R=\mathbb Z[\sqrt{-5}]$. Let $M=\langle 2,1+\sqrt{-5} \rangle$ an $R$-module. I am asked to show that $M \times M = \langle (2,1+\sqrt{-5}), ...
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Change of basis from Chebyshev to monomial basis for polynomials

I'm not that familiar with Chebyshev polynomials, so I hope I'm not too far off. Suppose that I have three order pairs $(x_0, f(x_0))$, $(x_1, f(x_1))$, and $(x_2, f(x_2))$ where $f : \mathbb{R} \to ...
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Form a basis for R^3? [on hold]

This is a homework problem and I need help on. Consider the matrix with the given vectors as its columns. Do (1, -1, 3), (-1, 5, 1), (1, -3, 1) form a basis for R^3?
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Proving a set is a basis for a subspace

The set $\{u_{1},u_{2}\cdots,u_{6}\}$ is a basis for a subspace $\mathcal{M}$ of $\mathbb{F}^{m}$ if and only if $\{u_{1}+u_{2},u_{2}+u_{3}\cdots,u_{6}+u_{1}\}$ is also a basis for $\mathcal{M}$. So ...
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2answers
17 views

Linear Algebra proof with column space

If $A$ and $B$ are two $m\times n$ matrices, then the column space of $A$ is contained in the column space of $B$ if and only if $A=BC$ for some $n\times n$ matrix $C$. So far I have that the rank of ...
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1answer
27 views

Show that Sine is not in the span of Cosine

Show that $\sin(x)$ is not in the span of $1$, $\cos(x)$, $\cos(2x)$, $\cos(3x)$, and $\cos(4x)$. Can I do this without Taylor series?
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Show that $F = \{a + b\sqrt{5} | a, b ∈ \mathbb Q\}$ is a field

Question: Show that $F = \{a + b\sqrt{5} | a, b ∈ \mathbb Q\}$ is a field under the operations - addition and multiplication where addition is given by: $(a + b\sqrt{5})+(c + d\sqrt{5}) = (a + ...
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8 views

why is the covariance matrix of a bekk model always positive definite?

The BEKK(1,1) model is given by: $$\Sigma_{t}=A_{0}A_{0}'+A_{1}a_{t-1}a_{t-1}'A_{1}'+B_{1}\Sigma_{t-1}B_{1}'$$ where $a_{t}$ are serially uncorrelated, zero mean innovations, $A_{0}$ is a lower ...
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1answer
15 views

Coordinates of a vector under a basis in a Hilbert space?

Given an arbitrary basis $\{m_1, \dots, m_n \}$of a Hilbert space $H$ (or just think it as $\mathbb R^n$, and I think the methods should be the same) with given inner product, how can we find the ...
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2answers
29 views

Show that a unique matrix exists for the coordinate vectors in a vector space

If $A=\{a_1,...,a_n\}$ and $B=\{b_1,...,b_n\}$ are two bases of a vector space $V$, there exists a unique matrix $M$ such that for any $f\in V$, $[f]_A=M[f]_B$. My textbook uses this theorem ...
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8 views

Find the eigenvector for an operator on a linear span

Let $V$ be the linear span of the functions $1,cos(x),sin(x)$. Let the operator $T$ on $V$ be given by the rule $Ty(x)=y(x+ \pi/4)$. Find the eigenvalues and eigenvectors of T in V. I know how to ...
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1answer
38 views

Proofs for $n$-dimensional vector spaces $V$

Suppose $V$ is an $n$-dimensional vector space. Prove that there is at most $n$ linearly independent elements in $V$. Prove that a set of $m<n$ element in $V$ cannot span $V$. I'm not really ...
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1answer
32 views

Matrix raised to a power

Find $A^n$ for $n = 1,2,...$. Does $A^n$ tend to a limit? $$A= \begin{pmatrix} 4/5 & 2/5 \\ 1/5 & 3/5 \end{pmatrix}$$ I found the eigenvalues $\lambda=1,2/5$ and the eigenvectors ...
2
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1answer
14 views

One Note about One to one and Surjective of linear functional [on hold]

I read a note that: if $ f \neq 0$ is a linear functional on H, then f is onto (surjective) and it is not one to one (injective) in general. Why this is true? i think it need advance ...