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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Linear transforms question

Let $T_s$ be the counterclockwise rotation about the positive y-axis through an angle \varphi. Write the standard matrix of $T_s$. I'm not entirely comfortable when questions present themselves ...
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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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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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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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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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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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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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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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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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Solve $y' = x + y$ 2 [on hold]

Find the no of positive integral solutions for the equation: (1/x) + (1/y) = 1/N! (the inverse of the factorial of N). Print a single integer which is the no of positive integral solutions for the ...
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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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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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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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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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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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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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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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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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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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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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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: ...
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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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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 ...
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How do you find L of LU for this LU factorization?

$$ \begin{bmatrix} 3 & -7 & -2 \\ -3 & 5 & 1 \\ 6 & -4 & 0 \\ \end{bmatrix} $$ The method my book gave me of doing this is: divide col1 ...
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1answer
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Finding a particular solution to the non-homogenous system

I have the following problem $\vec{x}^{'}(t)=\begin{pmatrix} 2 & -5\\1 & -2 \end{pmatrix}\vec{x} + \begin{pmatrix} \csc t\\ \sec t \end{pmatrix}$ Step 1) Find the Eigenvalues ...
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Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...
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Is it possible to diagonalize a singular matrix?

I have not seen anywhere written that it is impossible, but it seems impossible, so I want to check if I missed something. According to a theorem, an nxn matrix is diagonalizable if it has n ...
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Find the Axis of rotation of rotation matrix $K$ after solving $(K-I)v=0$

$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$ Find the axis of rotation for the rotation matrix $K$. This is from my previous thread click here ...
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Find a vector $t \in \{x,y,z\}$ with base $\{u, v, w\}$

I don't know how to find a vector $\vec t$ that will suffice the condition: $\vec t \in \{x,y,z\}$ with bases $\{u, v, w\}$ the given vectors are: $$ \begin{array}{rcrrrrrl} u &=& [ & ...
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Finding a normal to an ellipsoid

Let $E$ be an ellipsoid centered at $v = (x,y,z) \in \mathbb{R}^3$ and let $T:\mathbb{R}^3 \to \mathbb{R}^3 $ be a linear transformation which transforms $E$ to a sphere $S$ with a radius of length ...
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1answer
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$A$ = ($a_{ij}$) with $a_{12}$ = $1$; $a_{ij}$ = $0$ for all $(i,j)$ not eqal to $(1, 2)$

Let us consider an $n$x $n$ matrix $A$ = ($a_{ij}$) with $a_{12}$ = $1$; $a_{ij}$ = $0$ for all $(i,j)$ not eqal to $(1, 2)$. How to prove that there is no invertible matrix $P$ such that PA$P^{-1}$ ...
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Show the transformation matrix in relation to the canonical basis of the respective spaces.

$W = \{A \in M_{2\times2} (\mathbb{R}): A_{11} = A_{12}\text{ and }A_{22} = A_{21}\}$ is isomorphic to $P_1 (\mathbb{R})$. Show the transformation matrix in relation to the canonical basis of the ...
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How to setup linear transformation to find the image

Given $T:\mathbb R^3 \to \mathbb R^3$ such that $$T(1,1,1) = (5,0,-1),\ T(0,-1,2)=(-3,5,-1),\ T(1,0,1)=(1,1,0)$$ Find the indicated image $T(2,-1,1)$ The problem is that the text book only has a ...
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How to prove the controllability and observability?

The matrices $ C:=-(A^{\mathrm T}+QA^{-1}G)^{-1}(QA^{-1}GA^{-\mathrm T}Q+Q)(A+GA^{-\mathrm T}Q)^{-1}\\ D:=(A+GA^{-\mathrm T}Q)^{-1}\\ E:=-(A+GA^{-\mathrm T}Q)^{-1}(GA^{-\mathrm ...
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Normality of the product of a diagonal matrix and an SPD matrix?

I believe this to be true, but can't seem to prove it exactly: suppose $A$ is symmetric positive definite, and $D$ is a diagonal matrix. Then, $A$ is diagonal if $DA$ is normal for any diagonal ...
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Determinant of a matrix with symmetric positive definite block

In reviewing linear algebra for an exam, I encountered the following problem: Let $A \in \mathbb{R}^{n\times n}$ be symmetric positive definite. If $x$ is any nonzero vector, show that $$ ...
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determinant inequality, $AB=BA$, then $ \det(A^2+B^2)\ge \det(2AB) $

$A$ and $B$ are two $n\times n $ real matrices, $AB=BA$. Can we conclude that $$ \det \Big(A^2+B^2\Big)\ge \det(2AB) $$ is right? Well, the inequality is interesting. if $A,B$ are upper ...
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How can I compute pseudo determinant

Let A square n by n matrix and let b:=pseudo det of A And assume that A is diagonalizable and rkA=r Then what is pseudo det of AA^(t)??
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linear algebra, numerical linear algebra

Let A∈R^nxn have singular values σ1>=σ2>=...>=σn>0. Prove that minmag(A)=σn We have been working with calculating the SVD, and I know this has to do with the Sigma matrix of it. I am not sure where ...
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Kernel and composition of linear transforms. I don't want to multiply matrices, should be a theorem.

It's a 4x4 matrix, call it $A$, I could do $A\ne 0$, then $A^2=0$ (in this case) and just be able to stop there. But that involves multiplying matrices! Now I can get dim(kernel) easily, I just bolt ...