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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calculating the characteristic polynomial

I have the following matrix: $$A=\begin{pmatrix} -9 & 7 & 4 \\ -9 & 7 & 5\\ -8 & 6 & 2 \end{pmatrix}$$ And I need to find the characteristic polynomial so I use ...
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1answer
24 views

Uncoupled Linear System: Differential Equations

I'm trying to make sense of a problem I was given in class and I want to know if I am on the right track. The question is as follows: If $\vec{u}(t)$ and $\vec{v}(t)$ are solutions of the linear ...
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0answers
19 views

Probability that a Polynomial Has Specific Root When $y_i$'s are Not Random.

Imagine we have $\vec{x}=(x_1,...x_n)$ and two polynomials $P_1$ and $P_2$. Degree of $P_1$ is fixed $n-1$, but degree of $P_2$ can be at most $n-1$. $P_1$ has root $\beta$, where $\beta \leftarrow ...
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3answers
137 views

Prove that $\det(I-CD)=\det(I-DC) $

Let $C$ and $D$ be matrices such that $DC$ and $CD$ are square matrices of the same dimension. How can one prove that $\det(I-CD)=\det(I-DC)$? This is my approach to the question. I am not sure ...
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0answers
20 views

Measure change/similarity between two affine transformations

I have two affine transformations $A_1$ and $A_2$ consisting only of a rotation matrix $R_i$ and a translation vector $\overrightarrow{t_i}$ (all in 3D space): $$A_i = \left[ \begin{array}{ccc|c} \, ...
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2answers
32 views

Sum of three cross products is zero.

Let $u,v,w\in \mathbb R^3$. Prove $u \times( v \times w)+v \times( w \times u)+w \times( u \times v) =0$ I guess things would work out if I just expanded as a ton of products. Is there a better way?
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1answer
28 views

Characteristic polynomial above the complex numbers

I need to find the Characteristic polynomial of $\begin{pmatrix} i+1 & 0 & 0 \\ 0 & 3i-1 & 2-2i\\ 0 & 2-2i & 3i-1 \end{pmatrix}$ I know that there is not ...
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1answer
4 views

Solving Augmented Matrix Breaking Strict Triangle Form

I'm trying to solve the following system of equations: $ 3x_1 + 2x_2 + x_3 = 0\\ -2x_1 + x_2 -x_3 = 2\\ 2x_1 - x_2 + 2x_3 = -1 $ From which I'm using the augmented matrix: $$ \left[ ...
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0answers
43 views

Valid Vector Space Proof (given v + w = 0 prove w = -v)

I'm working through Serge Lang's 'Algebra' and my answers to the proof exercise differ from his but reach the same outcome. I'm not sure if my proofs are invalid or if they are a correct alternative ...
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1answer
36 views

how many solutions does a 3x3 linear systems have? [on hold]

I just want to know if 3x3 linear systems has no solution, one solution, or two solutions, three solutions or infinitely many.solutions
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2answers
33 views

proof-similar matrices have the same characteristic polynomial

I have this proof but did not understand the following step: $$ \det(xI - M^{-1} A M)= \det(M^{-1} xI M - M^{-1} A M)$$ The author said in the comments that it is due to "$xM$ commutes with the ...
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1answer
23 views

Selecting a basis such that the orientation is preserved

I need to map a polygon from a 3D plane to a 2-dimensional basis, do some processing, and project the result back to 3D. The vertices in the polygon is always ordered counterclockwise and this ...
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1answer
16 views

Determining intersection of kernel and range of a linear operator.

I was posed the following question : If T is a linear operator on vector space V and given that the kernel and range of T are disjoint, ie. they have only the zero vector in their intersection and T ...
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2answers
54 views

Proof for $V \cong V^{**}$

Theorem: Let $V$ be an vector space. Then the dual space of $V$'s dual space is canonically isomorphic to $V$. I am able to prove that $V$ is a subspace of $V^{**}$, the map ...
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1answer
23 views

How to find a $3\times 3$ matrix with a certain null space and certain column space

We were asked to Find a $3\times 3$ matrix whose null space is the $x$-axis and whose column space is the $yz$-plane. I was told that the answer is this, but I didn't understand why: ...
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0answers
23 views

In vector space why we not take 1.a=a.1 [on hold]

I solve many question which satisfies this condition.I want to answer this
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1answer
29 views

Is it possible for a matrix to have nullity different from its transpose?

Say I have a $2\times 3$ matrix with $\operatorname{rank}(A)=1$ and $\operatorname{nullity}(A)=2$, i.e. only one pivot point and two free variables in the solution set of this system because we have ...
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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.
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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 ...
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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 ...
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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 = ...
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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 ...
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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. ...
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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 $𝜎_{𝑖𝑗}+𝜎_{π‘–π‘˜}𝑀_{π‘˜π‘—}βˆ’π‘€_{π‘–π‘˜} 𝜎_{π‘˜π‘—}$ All the matrices in the equation ...
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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 ...
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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 ...
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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 βˆ’ 4|1i. (i) Find the matrix representation of ...
2
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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 ...
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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 ...
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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 ...
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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!
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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
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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?
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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 ...
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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 & ...
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0answers
29 views

Eigenvalues of Moore–Penrose Pseudo-Inverse of a Symmetric Matrix

I was wondering if there is any bound or inequality for the eigenvalues of Moore–Penrose pseudo-inverse of a real $n\times n$ symmetric matrix $A$ in terms of eigenvalues of $A$, namely $\lambda_i$'s ...
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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
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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 ...
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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 ...
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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} \\ ...
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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: ...
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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 ...
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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
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0answers
33 views

Find the extreme points of the below polyhedral sets

Find the extreme points of the below polyhedral sets: (a)$$ P =\{(x_1,x_2,x_3)|x_1 +x_2 +x_3 ≀1,x_1,x_2,x_3 β‰₯0\}.$$ (b)$$ P = \{(x_1, x_2, x_3 x_4|x_1+ x_2+ 0.5 x_ ≀ 1, x_1 x_2,x_3,x_4β‰₯ 0\}. $$ ...
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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 ...
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0answers
23 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 ...
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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
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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 ...
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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} ...
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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 $D$ ...