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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Finding matrix with respect to given bases

Given that A: \begin{matrix} a & b & c \\ d & e & f \\ \end{matrix} is a matrix of T : V -> W with bases G = {g1, g2, g3} and Q = {q1, q2}, respectively. Find the matrix of T ...
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Prove uniqueness of polar-coordinates $(R, \theta)$ up to angle $\theta+ 2 \pi$.

Prove uniqueness of polar-coordinates $(R, \theta)$ up to angle $\theta+ 2 \pi$. Suppose we have $(x,y) \in \mathbb R^2$. Then we can transform this point to polar-cordinates $(R, \theta)$ where ...
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26 views

If a 2x2 set of matrices are independent, do they also span M22?

I am looking for a basis in M22. So I need to get a linearly independent set that also spans the vector space. I have worked out how to tell independence, but I am stuck on the spanning requirement. ...
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1answer
11 views

Singular matrices over a commutative ring $R$, with a given adjoint matrix

First, I apologize if this is a duplicate question. I also must apologize if this has a trivial solution. This question has two parts: Let $R$ be a commutative ring with $1$, and let $F = R^n$ be a ...
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2answers
17 views

Diagonalization problems (eigenvalues and vectors)

I am trying to diagonalize the following matrices: $$A = \begin{pmatrix}0 & 1\\-1 & 2\end{pmatrix}\qquad B = \begin{pmatrix}1 & 2\\-1&-1\end{pmatrix}$$ For matrix $A$, I find an ...
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B is a basis of V implies B is a maximal linearly independent set of V

How would you prove that B is a basis of V implies B is a maximal linearly independent set of V?
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2answers
57 views

Determinant of m by m Matrix

How would you find the determinant of an $m \times m$ matrix which has $m$ as every diagonal entry and $-1$ as every non diagonal entry?
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3answers
62 views

Showing Orthogonality

How would I do this question..... I'm familiar with Gram-Schmidt and the basics but I have no idea how to do $a$ and $b$ in this question. Suppose $\{\vec x_1, \vec x_2, \vec x_3\}$ is an ...
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1answer
12 views

Matrices with Operator Norm 1

I believe the following claim is true, and I have a proof, but I'm still not sure. It seems like something I would have encountered by now if it were true. Suppose an matrix $A=(a_{ij})\in ...
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13 views

Given 2 matrices generate a reducible algebra, show they have a common eigenvector

Two matrices A, B generate an algebra.... the span of all words made with A and B... example of an element of the algebra: $A^kB^nA^m + I + B^sA^q$ etc... (exponents are all nonnegative). This algebra ...
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25 views

Find $P$ such $P^{-1}AP = kR$.

Let $ A=\begin{bmatrix} 3 & -5 \\ 1 & -1 \\ \end{bmatrix}$ Find P such that $P^{-1}AP = k R$ where $k \neq 0$ is a scalar and R an element of $SO(2)$ = {R element of ...
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15 views

Linear Algebra Question: Prove that no proper subset spans

I have to prove that "S is a basis for linear space L if and only if it is a minimal spanning set for L. In other words S is a basis for L if and only if S spans L and no proper subset of S spans ...
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Linear Algebra. Past Exam Question

Past Exam Question Help (a) Let $P_2(R)$ denote the vector space of real polynomial functions of degree less than or equal to two and let $B:= [p_0,p_1,p_2]$ denote the natural ordered basis for ...
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9 views

Resulting Covariance Matrix $\Sigma$' after reducing space along the primary eigenvector?

I am writing a quick & dirty C program to find the first three eigenvectors of a quite large system of points with 512 feature dimensions each. Data is all real. I find the first eigenvector ...
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8 views

A function $w(t)$ which satisfy $\int dt w(t)F[x](t)=c$

Consider a differentiable scalar function in two variables $F(x,t)$ for $x\in X$ and $t\in T$, then it can be viewed as an infinite family of scalar functions $\{F[x](t))\}_{x\in X}$. Are there any ...
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35 views

Find the axis of rotation of a rotation matrix by $INSPECTION$ (NOT by solving $Kv=v$)

$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$ Find the axis of rotation for the rotation matrix $K$ by INSPECTION. This is from my other thread ...
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25 views

Prove that if $v_1,v_2,…,v_r$ form a linearly independent set of vectors in $V$…

Let $S$ be a basis for an n-dimensional vector space $V$. Prove that if $v_1,v_2,...,v_r$ form a linearly independent set of vectors in $V$, then the coordinate vectors $(v_1)_S,(v_2)_S,...,(v_r)_S ...
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16 views

Linear transforms question

Let $T_s$ be the counter-clockwise rotation about the positive y-axis through an angle $\varphi$. Write the standard matrix of as $T_s$. I'm not entirely comfortable when questions present ...
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1answer
13 views

Similar Matrices Conditions:

Sorry this question was already asked but my english is not good. For matrix to be similar, does it have to have all of these properties or SOME of them? Same determinant Same Trace Same ...
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16 views

Prove that the $j$-th column of $AB$ is the product $Ab_j$

Prove that the $j$-th column of $AB$ is the product of $A$ and the $j$-th column of $B$ First of all, THIS IS NOT HOMEWORK. This was a homework. I can prove this using the fact that $e_j$ extracts ...
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Is $\|x_1\|^2 + 2\|x_2\|^2 > - 2\Re(ix_1\overline{x_2})$ for complex numbers $x_1,x_2$

This is the last piece I need for a proof for a homework problem. Could someone explain whether or not this inequality must hold?
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2answers
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Is there exist a linear map $T:\mathbb{R^2}\to\mathbb{R^3}$ such that $Range(T)=\{(x_1,x_2,x_3)∈ \mathbb{R}:x_1 + x_2 + x_3 = 0\}$?

Is there exist a linear map $T:\mathbb{R^2}\to\mathbb{R^3}$ such that $Range(T)=\{(x_1,x_2,x_3)∈ \mathbb{R}:x_1 + x_2 + x_3 = 0\}$? I do not understand what is actually I have to do here.I think it ...
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1answer
23 views

Is every invertible matrix over an algebraically closed field diagonalisable?

In $\Bbb{R}$ the only invertible matrices (I can think of) that are not diagonalisable are those which stand for a rotation, but in $\Bbb{C}$ this shouldn't be a problem anymore, since rotations can ...
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1answer
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help with simplifying this sum

Problem I need help with simplifying following sum: $$ 1 + \sum_{i=1}^{\infty}{\frac{1}{i!} * (-1)^i * a * (a + b)^{i-1}} $$ and can get the $a$ out to get $$ 1 + ...
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1answer
15 views

Are rank and determinantal rank the same over a PID?

Are the notions of rank and determinantal rank equivalent for an $m\times n$ matrix $A$ with entries in a principal ideal domain $D$? I'm specifically interested in the case $D=\mathbb{Z}$.
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24 views

If the rank$ (T)=1$ and Im $(T) \cap$ Ker$(T)$ is zero, show $T$ is diagnolizable.

The full question Let $T: V\rightarrow V$ be a linear operator. If the rank $(T)=1$ and Im$(T) \cap$ Ker$(T)=0$, show $T$ is diagonalizable. Alright so I've been trying my hand at this the past ...
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32 views

Algebraic mean problem

The Question is: $27pqr \geq (p+q+r)^3$ and $3p+4q+5r=12$, then what is the greatest value of $p^3+q^4+r^5$? How do i solve this problem? Im think harmonic mean has to be used along with geometric ...
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Integer solutions to the equation (1/x)+(1/y)=1/N! [on hold]

I am trying to find the number of positive integral solutions $(x,y)$ to the equation: $$\frac 1x + \frac 1y = \frac 1{N!}$$ where $N$ is a positive integer.
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20 views

Find the monic generator of and ideal.

Let $\mathbb{F}$ be a subfield of complex numbers, and let $$ A = \begin{bmatrix} 1 & -2 \\[0.1em] 0 & 3 \\[0.1em] \end{bmatrix} $$ Find the monic generator of the ideal ...
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Transform an almost positive definite matrix to positive definite matrix.

Matrix A (n by n) is constructed as followed. $A(i,i)= -\sum^{k=n}_{k=1}A(i,k), k \neq i. $ A is symmetric. All the off-diagonal elements are negative except two elements $A(p,q)$ and $A(q, p)$: ...
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3answers
33 views

Prove that if $\dim X'<\infty$ then $\dim X<\infty$

I have to prove that $\dim X'<\infty$ then $\dim X<\infty$ where $X$ is a normed vector space and $X'$ is a space of all linear and continuous functionals from $X$. How can I prove this? I ...
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Selection of matrix norms and numerical analysis

I have two formulae, $A_k^{-1}$ and $B_k^{-1}$, where $k=0,1,2,\cdots$ is the discrete time, $A_k,B_k\in\mathbb R^{n\times n}$. By theoretical analysis, it is already known that: $A_k$ monotonically ...
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61 views

A proof of $x^TAx=\mathrm{tr}(Axx^T)$

In this post here my answer was downvoted. Unfortunately, I cannot find the mistake and the downvoter did not comment. The question was to prove that $x^TAx=\mathrm{tr}(Axx^T)$. The argument I gave ...
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44 views

Show that if the set $\{v_1,v_2,v_3\}$ is linearly independent then so are all subsets. [on hold]

Show that if the set $\{v_1, v_2, v_3\}$ is linearly independent then so are $\{v_1, v_2\}$, $\{v_1,v_3\}$, $\{v_2,v_3\}$, $\{v_1\}$, $\{v_2\}$ and $\{v_3\}$. I don't even know what to start with. Do ...
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Does the cross section of $[-1,1]^n$ on a $k$-dimensional subspace always contains a rotated image of $[-1,1]^k$?

This question is inspired by a recent bounty question, but the two questions are different and solving this one, I believe, will not lead to an answer of that bounty question. Suppose $n>k\ge1$ ...
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Soccer betting in coding theory [on hold]

Problem 4.14(Ron Roth) A soccer betting form contains a list of 13 matches. Next to each listed match there are three fill-in boxes which correspond to the following three possible guesses: “first ...
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21 views

Determine if the following linear transformation is surjective or injective

Let $S \left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\right) = $ $\begin{pmatrix} x_1 & -2x_2 & x_3 & x_4\\ 2x_1 & - 4x_2 & -3x_3 & ...
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4answers
631 views

Is the determinant differentiable?

I was wondering, given an $n\times n$ square matrix with $n^2$ many entries, the function $\det:\left(a_1,a_2,\ldots,a_{n^2}\right)\to \textbf{R}$ which gives the determinant where $a_{k}$'s are the ...
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14 views

Collecting terms of a hard linear equation

I need to collect the $\Pr(\cdot)$ terms of the following expression: $\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left( 1-\theta \right) }\right) ^{m}}\left[ ...
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What's wrong with $\det(P) = -1$ : Change of variable for Quadric Forms ? [Kolman P552 8.7.25]

Would someone please explain "why $\det(P) = 1$ is required" and the general procedure of effecting this? Lay S7.2 didn't expound on this and neither does Kolman in S8.6-8.8. Identify the graph ...
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Matrix representations of Transformation with change of basis (Fraleigh Beauregard)

I'm having problems understanding section 7.2 of FB's Linear Algebra, 3rd edition, and I can't find the solution online since no specific name is given to the matrices. Sorry for the long ...
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1answer
28 views

What is a reducible algebra?

In my matrix analysis book, a set of complex matrices is said to be an "algebra" if 1)it is a subspace, 2)whenever A and B are members, so is AB. Then it uses the terms reducible and irreducible ...
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How to find the matrix of a transformation relative to standard basis?

Given $b_1=(-1,3)$ and $b_2=(1,-2)$ which make a basis for $\mathbb R^2.$ If $$ T(b_1) = 6b_1 + 7b_2 \quad\text{and}\quad T(b_2) = 3b_1 + 8b_2, $$ find the matrix of $T$ relative to the standard basis ...
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28 views

Calculating the Eigenvectors and Eigenvalues of this Matrix Polynomial

For the matrix $$ A=\begin{pmatrix} 1 & 1 & 2 \\ 0 & -2 & 0 \\ 0 & 2 & 3 \end{pmatrix} $$ How are the eigenvalues and eigenvectors of the following matrices calculated? ...
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Impact of the transformation matrix distribution on linear transformation

Let $X$ be a $m\times n$ ($m$: number of records, and $n$: number of attributes) normalized dataset (between $0$ and $1$). Denote $Y=XR$, where $R$ is an $n\times p$ matrix, and $p<n$. I understand ...
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Linear Algebra - Basis and geometry

Given the set $W = \{q,w,e,r\}\ |\ w-e+r=0] \subset\mathbb{R}^4$ i) Find the Basis and describe the Set W geometrically. (Assume for now I have proven it is a subspace of $\mathbb{R}^4$ My attempt: ...
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2answers
30 views

Proof that if we add a vector to a linearly dependent set of vectors in a vector space $V$, then the new set of vectors is still linearly dependent

Prove that if $S=\{v_1, v_2, v_3\}$ is a linearly dependent set of vectors in a vector space $V$, and $v_4$ is any vector in $V$ that is not in $S$, then $\{v_1, v_2, v_3, v_4\}$ is also linearly ...
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3answers
47 views

Explain why $S$ is not a basis for $\mathbb{R}^3$

Explain why $S$ is not a basis for $\mathbb{R}^3$ $S=\{(1, 3, 0),(4, 1, 2),(-2, 5, -2)\}$ I set this equal to an arbitrary vector $\mathbf{x} = (x_1, x_2, x_3)$ After solving I got the matrix: ...
2
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3answers
63 views

Find dimension of ℒ $(V)$ and polynomial that brings every linear transformation to $0$

Here's the prompt: Let V be a vector space of finite dimensions $n$ over the field $\mathbb{F}$, and let $\tau \in$ ℒ $(V)$. What is the dimension of ℒ $(V)$ as a vector space over $\mathbb{F}$? With ...
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1answer
26 views

Cannot find eigenvectors

How can I find eigenvectors of the following matrix? $$ \begin{matrix} 4 & 0 \\ 0 & 1 \\ \end{matrix} $$ Systematic approach would be: 1. Finding eigenvalues ...