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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2
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3answers
75 views

If square matrix A satisfying $A^2-4A+4I=0$ does it follow that A is diagonizable?

I am given the following statement and asked to determine whether it is true or false: If A is a n x n matrix, and $A^2-4A+4I=0$, then A is diagonizable. Any help is appreciated, thank you.
3
votes
1answer
31 views

How many numbers between 1 and 10000, inclusive, are multiples of 12 or 20?

I calculated the multiples of 12 and multiples of 20, 833 and 500 respectively. Now I calculated the multiples of 12 * 20 = 240,and as a result have 41. The solution would be 833 + 500-41 = 1292 ...
0
votes
2answers
33 views

Show that the full null space of the matrix A and its column space in the plane 2x+2y - z = 0

Show that the full null space of the matrix A = $\begin{bmatrix} 0&1&5\\ 1&0&0 \\ 2&2&10 \end{bmatrix}$ is the line $\lambda$(0.-5,1), $\lambda \in \mathbb R^3$ and its ...
3
votes
0answers
32 views

Do I have the correct mental map for adjoint operators for inner product spaces?

Let $X$, $Y$ be finite dimensional inner product spaces, let $A: X \to Y$ be a linear operator, let $A^*: Y \to X$ be the adjoint operator to the linear operator, defined using $<y, Ax>_Y = ...
0
votes
1answer
24 views

about scaling property of proximal operator

If the proximal operator of $f(x)$ is $\text{prox}_{\lambda f}(x)$, what about $cf(x)$ and $f(cx)$, c is a scalar. For example, If $f(x) = ||x||_{1}$, $x \in \mathbb{R}^{n}$, how about the proximal ...
2
votes
0answers
31 views

Linear map is diagonalizable iff its adjoint is diagonalizable

Problem Let $V$ be a finite inner product space and let $T:V \to V$ be a linear transformation. Prove that $T$ is diagonalizable if and only if the adjoint transformation $T^{*}$ is diagonalizable. ...
2
votes
2answers
39 views

Index notation interpretation for matrices

I want to understand the how to interpret the matrices which are represented by index notation. Here is my matrix $๐œŽ_{๐‘–๐‘—}+๐œŽ_{๐‘–๐‘˜}๐‘ค_{๐‘˜๐‘—}โˆ’๐‘ค_{๐‘–๐‘˜} ๐œŽ_{๐‘˜๐‘—}$ All the matrices in the equation ...
1
vote
2answers
38 views

Explicit example of a basis of invertibles for $n\times n$ matrices

Using a topological (+linear algebra) argument, one can establish the existence of a basis spanning any square matrix using invertible matrices ( $span(GL_n (\Bbb{R}))=\mathcal{M}_n (\Bbb{R}) $). But ...
1
vote
0answers
35 views

Least squares solutions of the linear system

I'm doing problems from old exams, and my solutions don't add up with the professor's solution. The problem is as followed: Find all least squares solutions of the linear system. I checked my ...
-5
votes
0answers
21 views

mathematics representation on operators [on hold]

(a) Let A be an operator on C and |1i = 2 which has the following action on the canonical basis vectors |0i = 0 : 1 A|0i = 2|0i + 3|1i A|1i = 1|0i โˆ’ 4|1i. (i) Find the matrix representation of ...
2
votes
1answer
38 views

Orthogonality v. Perpendicularity

In Intro to Linear Algebra (my class) two vectors are defined to be orthogonal if their dot product is zero. And the dot product of two $n$-vectors $\vec a\cdot\vec b=0$ means that the two vectors are ...
1
vote
1answer
50 views

Intuition: why distinct eigenvalues -> linearly independent eigenvectors?

Suppose you have an n x n matrix with n distinct (not repeated) eigenvalues. There is a theorem telling us that the eigenvectors corresponding to these eigenvalues must be linearly independent. I can ...
0
votes
0answers
13 views

Derivating the normal vector from a line equation

$$g_a: \binom{x}{y}=\binom{x_g}{y_g}+\lambda\binom{a_{g_a}}{b_{g_a}}$$ $$g_b: a_{g_b}x+b_{g_b}y=c$$ Explain why: $$\vec{n_{g_a}}=\binom{a_{g_b}}{b_{g_b}}$$ Where ...
0
votes
3answers
33 views

matrix times its transpose equals minus identity

What would be a good example for a $n\times n$ matrix such that $A^{T}A=-I$? It would be better if you can give a matrix which has a well-known name (like "rotation matrix" etc). Thanks!
0
votes
1answer
34 views

Find a $4\times 3$ matrix to make a $3 \times 4$ matrix invertible [duplicate]

The matrix $P$ is 3x4 given. Is there a 4x3 matrix $Q$ such that the product of $QP$ is invertible? If this can't happen can someone explain why?Thanks. Matrix P
1
vote
3answers
57 views

Can the product of an $4\times 3$ matrix and a $3\times 4$ matrix be invertible?

I want to find the inverse of the product of $2$ non-square matrices, but is this even possible?
0
votes
0answers
25 views

Find a relation between a,b and c

$ a,b,c\in \Bbb R$ $2x_1+2x_2+3x_3=a$ $3x_1-x_2+5x_3=b$ $x_1-3x_2+2x_3=c$ if a,b and c is a solution of this linear equation system find the relation between a,b and c I dont understand the ...
2
votes
2answers
43 views

Computing matrix exponential of non-diagonalizable 2x2 matrix

Compute $e^M$ where $M=\begin{bmatrix}8 & -1\\4 & 4\end{bmatrix}$ Because M is not diagonalizable i try to use Jordan decomposition so i find the Jordan matrix to be $J=\begin{bmatrix}6 & ...
1
vote
0answers
29 views

Eigenvalues of Mooreโ€“Penrose Pseudo-Inverse of a Symmetric Matrix

I was wondering if there is any bound or inequality for the eigenvalues of Mooreโ€“Penrose pseudo-inverse of a real $n\times n$ symmetric matrix $A$ in terms of eigenvalues of $A$, namely $\lambda_i$'s ...
0
votes
1answer
28 views

Union of subspace

Q. Say U and W are subspaces of a a finite dimensional vector space V (over the field of real numbers). Let S be the set-theoretical union of U and W. Which of the following statements is true: a) ...
2
votes
1answer
57 views

For Which Value The Matrix is Diagonalizable?

For which values of $a$ the matrix $\left(\begin{array}{ccc} 2 & 0 & 0 \\ 2 & 2 & a \\ 2 & 2 & 2 \end{array}\right)$ is diagonalizable: above $\mathbb{R}$ above ...
0
votes
0answers
10 views

How to rescale parameters?

First of all, I am a maths newby and never got any education on rescaling parameters on whatsoever. The knowledge that I have is based on what I know from mathematical research papers and as ...
0
votes
2answers
25 views

How to solve singular linear algebra problem for just one element

I am right now confronted with the linear algebra problem: \begin{equation} \begin{pmatrix} m_{11} & m_{12} & m_{13} \\ m_{12} & m_{22} & m_{23} \\ m_{13} & m_{23} & m_{33} \\ ...
0
votes
0answers
34 views

Power of linear transformation

Let $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be defined as: $$T\left(\begin{array}{c} x \\ y \end{array}\right)=\left(\begin{array}{c} 5\,y+13\,x \\ -12\,y-30\,x \end{array}\right)$$ Find: ...
2
votes
1answer
55 views

Interpolating a Polynomial with a Subset of Interpolation Points

Consider we has a polynomial $P=(x-\beta)g(x)$, where $\beta \leftarrow \mathbb{Z}_p$, $p$ is a large prime, and $g(x)$ is a non-zero polynomial. Here degree of $P$ is fixed $n$. We evaluate $P$ at ...
1
vote
1answer
39 views

How to find the inverese of the matrix in modulo 5

View the matrix I know how to do the inverse and think I know the right answer in modulo 5 but need to make sure thanks
-1
votes
0answers
33 views

Find the extreme points of the below polyhedral sets

Find the extreme points of the below polyhedral sets: (a)$$ P =\{(x_1,x_2,x_3)|x_1 +x_2 +x_3 โ‰ค1,x_1,x_2,x_3 โ‰ฅ0\}.$$ (b)$$ P = \{(x_1, x_2, x_3 x_4|x_1+ x_2+ 0.5 x_ โ‰ค 1, x_1 x_2,x_3,x_4โ‰ฅ 0\}. $$ ...
1
vote
2answers
63 views

What does ||u|| mean?

What does $\left\Vert \mathbf{u}\right\Vert$ mean in this equation? How would this equation be performed? I'm extremely terrible in discrete mathematics and a simplistic answer would be ideal. (Don't ...
0
votes
0answers
22 views

If $C^H$ is the conjugate transpose of $C$ then what is meant by $C^{-H}$?

If $C^H$ is the conjugate transpose of $C$, i.e., $C^H=\overline{C^T}$ then what is meant by $C^{-H}$?. Assume that $C$ is a square matrix. I can't find a definition for this anywhere?. Can anybody ...
0
votes
1answer
25 views

Linear Systems: Differential Equations

The book being used for the course is Differential Equations and Dynamical Systems by Lawrence Perko. The question is as follows: Find the general solution of the linear system ...
4
votes
1answer
54 views

What mathematics topics pertain more towards applied mathematics?

I'm entering my second year of undergrad (majoring in mathematics), and I've found that I am really bad at Linear Algebra, but very good at Calculus and Differential Equations. I'm hoping to venture ...
1
vote
2answers
40 views

How can you expand the adjoint of a matrix into a polynomial with matrix coefficients?

This book contains an algorithm which claims that a matrix $sI - A$, where $A$ is some $n \times n $ square matrix and $s$ a variable can be expanded into $$adj(sI - A) = K_0 s^{n-1} + K_1 s^{n-2} ...
0
votes
4answers
60 views

Is it possible to find the criminal with graph-theoretic methods?

I've been presented to a problem: Someone commited a crime. When interrogated, the people, named $G,m,M,J,D$ argued: $G:$ It wasn't $D$; It was $M$. $m:$ It wasn't $M$; It wasn't $D$ ...
2
votes
2answers
71 views

How to derive this matrix equation

$$ \left< Z,X-L-S \right> \quad +\quad \frac { r }{ 2 } \left\| X-L-S \right\|_F^2 \quad =\quad \frac { r }{ 2 } { \left\| L-\left( X-S+\frac {Z}{r} \right) \right\| }_F^2 $$ I think $ ...
3
votes
4answers
19 views

An equation to represent all vector solutions to a system of equations with infinite solutions

If both $x$ and $y$ are solutions to a system of linear equations with infinite solutions then $$z = ฮฑx + (1 โˆ’ฮฑ)y$$ is also a solution for any real ฮฑ. I'm having some trouble understanding this. ...
0
votes
1answer
26 views

General Element of U(4)

Relating back to previous question about how to write a general element of $U(2)$, I am now wondering about how to write a general element of $U(4)$. Define $\Gamma_{(i,j)}:=\sigma_i\otimes\sigma_j$ ...
0
votes
0answers
34 views

Independence of the choice of base for the differential.

Let $f:\mathbb{R}^m\longrightarrow \mathbb{R}^n$ be a mapping, defined by differentiable functions which, generally speaking, are non-linear and map zero into zero: ...
0
votes
1answer
64 views
+50

A question about an infinite sequence of elementary row operations

Do there exist matrices $A$ and $B$ such that $B$ can be transformed into $A$ only if an infinite number of elementary row operations are performed on $B$? "What can we multiply the top equation by ...
3
votes
2answers
36 views

Unique Linear Map- Linear Algebra

Let $E = {e_1, . . . , e_n}$ be a basis for $\mathbb{R}^n$ , and let $v_1, . . . , v_n$ be arbitrary vectors in $\mathbb{R}^m$. Prove that there is a unique linear map $T : \mathbb{R}^n \rightarrow ...
1
vote
1answer
54 views
+200

3D projection coordinates onto 2D plane to determine transformation matrix?

I'm not sure if there is an actual solution to this problem or not, but thought I would give it a shot here to see if anyone has any ideas. So here goes: I basically have three vertices of a rigid ...
1
vote
1answer
28 views

Row equivalence implies independent columns?

I need to prove that "given" two matrices are row equivalent, a set of columns of the first matrix are linearly independent iff the corresponding columns of the second matrix are linearly independent. ...
0
votes
2answers
45 views

Eigenspace of a linear transformation

Let $V=\mathbb{R}_2[x]$ and $T:V\rightarrow V$ be defined as follow: $T\left(6\,x^2-x+2 \right )=30\,x^2+10\,x+43$ $T\left(5\,x+3 \right )=20\,x+12$ $T\left(5\,x-1 \right ...
1
vote
1answer
13 views

Linear transformation with matrices in base

Consider the vector space of real $2 x 2$ matrices and take as base $\{{E_{11},E_{12},E_{21},E_{22}}\}$. Where $E_{ij}$ represents the matrix with a $1$ in the $i$-th row and $j$-th column and the ...
0
votes
2answers
25 views

Explaining the number of solutions of a matrix (linear algebra)

Let A be a m-by-n matrix and consider the system of linear equations $Ax=b$. We know that this system can either have no solution, exactly one solution or infinitely many solutions. Using linearity ...
2
votes
1answer
34 views

exercise Question-29 from contemporary abstract algebra [on hold]

Consider the element A=(1101) in SL(2,R) what is the order of A? If we view A=(1101) as a member of SL (2,Zp), what is the order of A
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1answer
25 views

Least squares w.r.t. different basis

I am looking to solve the following equation, where $A$ is a diagonal matrix: $$\min_x\ (Lx - f)^T A (Lx - f)$$ which I recognize to be similar to least squares, but then with respect to a scaling ...
3
votes
2answers
40 views

What is an Eigenbasis and how do I calculate it with the information below.

I have the matrix $$A = \begin{bmatrix} 4 & 2 & 2\\ 2 & 4 & 2\\ 2 & 2 & 4 \end{bmatrix}$$ I've calculated the Eigenvalues and Eigenvectors as follows with help in a previous ...
0
votes
0answers
24 views

About Carl Meyer's matrix analysis

I have taught some part of it to myself when i was an engineering student. but now i changed my major to the pure math so now i am studying math as an undergraduate student. i thought the book is ...
0
votes
0answers
17 views

Why do the diagonals of a matrix have to be greater than 0 for the matrix to be positive definite? [duplicate]

Why do the diagonals of a matrix have to be greater than 0 for the matrix to be positive definite? Please provide an example (with numbers if possible).
0
votes
2answers
24 views

Incomplete Question? - Ratio

I have encountered the following question in the famous "5LB book of GRE" book If the zoo currently has 80 total birds, what is the samllest number of birds that could be added such that atleast 20% ...