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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Semplify $\det\left(D+M+A\right)$.

Let $D$, $M$, $A$, $n\times n$ matrices, with $n\in\mathbb N$. $D$ is a diagonal matrix, $M$ with elements all equal to $k\in\mathbb R$, $A$ is an antisymmetric matrix. Is possible to calculate ...
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3answers
24 views

What does it mean to evaluate a polynomial at a linear map, e.g. $p(T)$ where $p(x)$ is a polynomial?

I'm learning about the Cayley-Hamilton Theorem and the Primary Decomposition Theorem, and have trouble understanding what the notations mean. What does $\chi_T(T)$ mean? ($\chi_T(x)$ is the ...
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1answer
16 views

What can I assume, when given a matrix with information about its eigenvalues but not its action?

Basically, I've had to use linearity a couple of times yesterday and today, in order to write up a few proofs. But I notice that I am only given information such as positivity conditions and ...
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1answer
28 views

T-invariant subspaces

I'm studying for a qualifying exam, and I'd really appreciate some help on the following questions. Let p be a prime integer, $F=\mathbb{Z/pZ}$, V a vector space over F, and $T\in\mathscr{L}(V)$. ...
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21 views

Eigenspaces of $Y$.

Given an even dimensional real vector space and a complex structure $Y$, why is its complexification the direct sum of $\text{ker}(X + iY)$ and $\text{ker}(X - iY)$?
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3answers
81 views

If det $A = 0$ and $\det B \neq 0$ then show that $abc = -1$

This has been hurting my head for a while now.... If $$ \det\begin{bmatrix}a&a^2&1+a^3\\b&b^2&1+b^3\\c&c^2&1+c^3\end{bmatrix}=0 $$ And $$ ...
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13 views

Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions: $$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$ and $$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$ such ...
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20 views

A subgroup of special linear group

Does anybody know if the subgroup of diagonal and antidiagonal matrices of $SL(n,F)$ has been given a particular name? By $SL(n,F)$ I mean $n \times n$ matrices over a field $F$ with determinant 1. ...
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18 views

Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
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42 views

Basis for a eigenspace (multiple choice problem)

The following (multiple choice) problem is from a test review. For the given matrix $A$, find a basis for the corresponding eigenspace for the given eigenvalue. $$A = \begin{bmatrix}1 & 6 ...
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23 views

Intuition behind Equation for point on a line between two solutions for linear equations

I am trying to solve some basic linear algebra problem. The book says a point on a line between two solutions ( for linear equations) is given by $c (sol1) + (1-c) (sol2)$ I don't understand this. ...
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25 views

Understanding Householder Transformations

I've been thinking about Householder transformation for the past few days and one point appears to be escaping my insight. I hope that the following description helps someone correct my understanding ...
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1answer
23 views

Find the maximum value of this form

Let $A,B$ be $n\times n$ real symmetric matrices such that $B$ is positive definite. Show that $G$ defined below attains a maximum value at an eigenvector related to $A$ and $B$. Also find the ...
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8answers
199 views

Why is this set of polynomials linearly dependent?

$$1 + 2t+ t^2, 3-9t^2,1 + 4t + 5t^2$$ (A) Linearly dependent or (B) Linearly independent The answer is A from the answer key. This is a test review. I don't see that either ...
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0answers
12 views

Lagrangian complement in symplectic vector space

Let $(V,\omega)$ be a vector space over $K$ together with symplectic form $\omega : V \times V \rightarrow K$. Let $U \subseteq V$ be a Lagrangian subspace (in other words, $U = U^{\perp}$). I want to ...
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3answers
29 views

Prove {$v_1,v_2,w$} is a basis for vector space _V_

A problem from my textbook states: Let {$v_1,v_2,v_3$} be a basis for a vector space $V$. Prove that, if $w$ is not in $sp(v_1,v_2)$, then {$v_1,v_2,w$} is also a basis for $V$. Assume ...
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1answer
33 views

Particular form of the kernel of a positive matrix

Let $\mathbf L$ be a positive ($L_{ij} \geqslant 0$) $k \times n$ matrix with $k < n$. I'm looking for a matrix $\mathbf H \in \mathbb R^{n \times (n-k)}$ with two properties: columns of ...
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4answers
55 views

If $V$ is a vector space, then, proving that…

I have a big problem with this problem... : If $V_m(\mathbb{R})$ is a vector space whose dimension is "$m$" then Proving that "$m$" is even number if and only if exist an endomorphism $J$ of ...
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1answer
16 views

Reducing a Matrix with Guass-Jordan elimination

I am doing some homework, but I am stuck on this problem. I have to take a (I'd upload an image, but being new, I can't): -x+y=-2 -3x+2z=13 2x-2z=-6 Here's a picture of my work(I can only link ...
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2answers
35 views

Problem: linear form

I need help with this problem: we are on $\mathbb{R}^2$ and we consider two non-null vectors, $ \vec{v}, \ \vec{w}$, and a non-trivial linear form $\phi\in{\mathbb{R}^2}^*$. Now if ...
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3answers
23 views

Problem about planes

Say we have $2x+3y+3z=0$ which is a plane. Does that plane have infinite dimensions (it is a 2D "object" — forgive me as I am not a mathematician — but each side has infinite length) or is it just the ...
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0answers
27 views

Are all 7-dimensional cross products isomorphic?

Let $\hspace{.04 in}f$ and $g$ be bilinear maps on $\mathbb{R}^7$ which satisfy the orthogonality and magnitude conditions. Does there necessarily exist a linear map $\: \phi : \mathbb{R}^7 \to ...
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2answers
30 views

Proving that $h$ is an automorphism if $2h^2+4h+2I=0$

I dont know how i can solve this problem. You have one vector space $V$ with a linear transformation: $h: V \to V$. So we have an endomorphism. The question is to show that if $h$ satisfies ...
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2answers
33 views

Distance between point and plane - why use the dot product?

So according to this, the signed distance between a point and a plane will be the dot product of the plane's normal vector (does it have to be a unit vector?) and the point-in-plane minus the point ...
2
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2answers
32 views

Dimension and basis of a quotient space

I'm having some problems understanding this: $$V = \mathbb R^3\text{ and }W = \{(x,y,z) \mid x+y+z=0\}$$ So I want $V/W$ and a basis to it. $$\dim V = 3$$ $$\dim W = 2$$ $$\dim V/W = 1$$ But a ...
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0answers
7 views

Find a specific intersection point of line and Fourier series using Newton–Raphson method when graphs have more then one intersection points

I need to find an intersection point of two graphs in polar coordinates. First is defined by a simple line $y=kx+b$. Second — by Fourier series $$ r=r_{0} + \sum [a_{i}\cos(i\phi) + b_{i}\sin(i\phi)] ...
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15 views

What is pseudospectra of matrix polynomials? .

What is pseudo spectra of matrix polynomials? Please guide me with some example or some reference regarding it. Thank You!
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2answers
43 views

For what values of $a$, $b$, and $c$ the above system has: One solution. Infinitely many solutions. No solutions.

I am stuck with this now, I tried reducing the matrix to row echelon form, but it gets a bit hard. Is there not a simpler way? The system is: \begin{align*} a x + b y − 3 z &= −3\\ −2 x − b y + ...
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5answers
163 views

Find the dimension of a vector subspace

I'm doing a problem on finding the dimension of a linear subspace, more specifically if $\:$ {$f \in \mathcal P_n(\mathbf F): f(1)=0, f'(2)=0$} is a subspace of $P_n$, what is this dimension of ...
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1answer
32 views

Eigenvector and eigenvalue of the differential operator $L(x)=x''+3x'-4x$

This is a follow up question to this one. Just to summarize. I want to find the kernel, eigenvalues and eigenvectors of the differential operator: $$L(x):=x''+3x'-4x$$ In other words I want to find ...
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0answers
13 views

Properties or solution of C=I+wCw^T matrix equation?

In a project, I came to the following matrix problem: $$C_1=wC$$ $$C=I+wCw^\dagger$$ Where the unknown matrices are $C$, which is hermitian positive definite, and $w$, general not hermitian, no ...
2
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3answers
131 views

Find a constant so matrix is invertible

I am doing some exercises from my Linear Algebra textbook and i have come across an exercise which I don't quite understand. Every exercise is graded with numbers from [1] to [5]. [1] is meant to be ...
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2answers
22 views

Span and linear independence. Test review; not homework.

The following is a question on my exam review. The answer (from the answer key) is C, but why? Here is my analysis: The set does not span $R^3$ because there is not a pivot in every row of ...
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1answer
36 views

What is the relation between rank of a matrix, its eigenvalues and eigenvectors

I am quite confused about this. I know that zero eigenvalue means that null space has non zero dimension. And that the rank of matrix is not the whole space. But is the number of distinct eigenvalues ...
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2answers
34 views

The set of all vectors of the form $(a-4b,5,4a+b,-a-b)$ is not a vector space

This isn't homework, it's test review. Will someone tell me if my analysis is correct? The answer is D (not a vector space) because the zero vector is not possible. Setting $a = 0$ and $b = 0$ makes ...
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1answer
47 views

Consequences of the positivity condition $v^t A v > 0$ for the eigenvalues of $A$

Let $A$ be an $n \times n$ symmetric real matrix with n distinct eigenvalues $\lambda_1 , ... , \lambda_n$. a) Suppose $v^t(Av)$>0 for all v in $R^n$, v$\ne$0. Show that all $\lambda_i$ are positive ...
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1answer
30 views

Example of where the sum of a subspace and its orthogonal complement is not the original vector space?

Suppose $\mathbb{F}$ is an arbitrary field and let $W$ be a subspace of $\mathbb{F}^n$. $W^\perp$ can be defined in exactly the same way as in the real case. Show by example that it isn't ...
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2answers
46 views

Does a basis for an $n$-dimensional vector space have to have $n$ vectors?

For example, for $\mathbb{R}^n$, if I form a basis, do I need at least $n$ vectors in my basis set? In other words, can I form a basis for $\mathbb{R}^n$ by using only $n-1$ or less number of ...
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1answer
23 views

Transition matrix question,

In diagonalizing a matrix A, we use a matrix S, which consists of eigenvectors of A. To compute S, we simply take each eigenvector and write it as a linear combination of the standard basis. So if ...
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0answers
18 views

Inverting change of basis matrices to get back the original coordinate vector

Let $B=\{b_{1},b_{2}\}$ and $C=\{c_{1},c_{2}\}$ be bases for a vector space $V$, and suppose $b_{1}=-2c_{1}+4c_{2}$ and $b_{2}=3c_{1}-6c_{2}$. a. Find the change of coordinates matrix from $B$ to ...
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How do I maximize each value, while having them be as far apart as possible? [on hold]

I have three values V, S, and A. They sum to 1, and are all greater than 0. How do I maximize each value while having them be as far apart as possible? That is, I'd like V to be clearly greater than ...
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3answers
171 views

Find this Determinant

I have to find this determinant, call it $D$ \begin{vmatrix} \frac12 & \frac1{3}& \frac1{4} & \dots & \frac1{n+1} \\ \frac1{3} & \frac14 & \frac15 & \dots & ...
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1answer
33 views

How do I determine the weight to assign to each bucket?

Someone will answer a series of questions and will mark each important (I), very important (V), or extremely important (E). I'll then match their answers with answers given by everyone else, compute ...
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1answer
14 views

Finding change of basis matrix when given two bases as a set of matrices

Find the change of basis matrix between the following bases: $\alpha = \left\{ \begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}, \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix}, ...
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3answers
41 views

Book for Linear Algebra and Matrix

my major is Electrical Engineering and I am new in linear algebra and I need to be familiar with matrix theory deeply because of my research topic which is Image Processing. But, I do not know from ...
2
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1answer
16 views

Find intersection of non-parallell planes without further assumption on their normals

Finding the intersection line between two planes is basic linear algebra but is it possible to find one formula, without having to dealing with different cases? Example: $$ \left\{ \begin{aligned} ...
2
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3answers
18 views

Determining the formula for a linear map

Determine the formula for the following linear map: $L : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ with $L(1,2) = (0,-1)$ and $L(-1,-1) = (2,1)$. Attempt at solution: On the basis of these examples I ...
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1answer
43 views

When is the matrix $\mathbf{Y}=\mathbf{A}\mathbf{x}\mathbf{x}^{T}\mathbf{A}^{T}$ a symmetric matrix?

let $\mathbf{A}\in\mathbb{R}^{m\times n}$ and $\mathbf{x}\in\mathbb{R}^{n\times 1}$. \begin{equation} \mathbf{Y}=\mathbf{A}\mathbf{x}\mathbf{x}^{T}\mathbf{A}^{T} \end{equation} Can we say that ...
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1answer
10 views

Sesquilinear Forms: Polarization

This thread is only Q&A.* Given a Hilbert space $\mathcal{H}$. Consider the transforms: $$q[\varphi]:=s(\varphi,\varphi)\quad s(\varphi,\psi):=\frac{1}{4}\sum_{\alpha=0\ldots3}i^\alpha ...