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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Characteristic polynomial and minimal polynomial

Suppose that $x^{2}(x+2)$ is the minimal polynomial of a matrix A of order $5\times 5$. Specify all possible characteristic polynomials for this matrix? In each case discuss whether the matrix is ...
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Linear Algebra, Vector spaces, Complex.

$W = {\rm span}\{(1,3,4),(2,5,1)\}$, $U = {\rm span}\{(1,1,2),(2,2,1)\}$. Find a set that spans $U\cup W$. Also, another question about $\Bbb C$: If $z_1 \cdot z_2 \ne -1$, $|z_1|=|z_2|=1$, is ...
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2answers
35 views

Proving Eigenvalue squared is Eigenvalue of $A^2$

The question is: Prove that if $\lambda$ is an eigenvalue of a matrix A with corresponding eigenvector x, then $\lambda^2$ is an eigenvalue of $A^2$ with corresponding eigenvector x. I assume I need ...
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28 views

Reduced row echelon form with variables

I'm new to this, but if I have the matrix \begin{equation} A= \begin{bmatrix}1&2&3&1\\2&1&1&x^2+x \\ 3&6&x&x-6\end{bmatrix}\end{equation} and if I want to use the ...
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How do I find some distinct vectors u, v, w such that span({u, v}) = span({v, w})?

$span(${u, v}$)$ is a set that contains all the linear combinations $au + bv$ where $a$ and $b \in \mathbb R$. $span(${v, w}$)$ is a set that contains all the linear combinations $av + bw$ where $a$ ...
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Radial neuron teaching

Hello i have a task to write programm for teaching radial neuron with 3 inputs, i can't find some information about it, i find a lot of info about teaching netowork. I can't undestand what algorithm i ...
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23 views

Solving a system in 3 variables problem?

I need an answer for this problem, thanks in advance for the help. Find $x$, $y$, and $z$ from the problem below. \begin{eqnarray*} -2x + 1 &=& 5 \\ \\ 2x + 3y - 4z &=& 7 \\ \\ 3x ...
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19 views

Collinearity in geometry

Let S be the intersection of diagonals in a cyclic quadrilateral. Let p be a circumcircle of a triangle ABS and it intersects BC in M and q is a circumcircle of a triangle ADS and q intersects CD in ...
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24 views

Solving Absolute Value Inequalities “or” or “and” statement deciphering help?

When solving equations such as $|2x-4|>-12$ and $|3x-4|<9$ how can one tell if it is an "or" statement or an "and" statement? ${}$
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14 views

Dual norm of the k-norm

I need to show that the dual norm of the k-norm on $R^n$ and $C^n$ is $\lVert x\rVert^D_{[k]}=max\{\frac{1}{k} \| x\|_1,\lVert x\rVert_{\infty}\}$ The k-norm is defined as the sum of the k largest ...
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Connecting inner products and the trace of a matrix

Hello, I'm currently trying to work through this question. However, I am struggling with understanding what I should do. I am aware of all of the definitions used throughout, however I cannot link ...
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A Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues - Proof Strategy [Lay P397 Thm 7.1.3c]

Would someone please explain the proof strategy at Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues? I brook the algebra so I'm not asking about ...
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19 views

Show $p_1,p_2,p_3$ lies on a straight line in $\mathbb R^3$ if and only if $p_3 -p_1$ and $p_2-p_1$ are linearly dependent.

Let $p_i=(a_i,b_i,c_i)^T$ for $i = 1,2,3$ denote three different elements in $\mathbb R^3$. Show $p_1,p_2,p_3$ lies on a straight line in $\mathbb R^3$ if and only if $p_3 -p_1$ and $p_2-p_1$ are ...
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3 views

Dual Lefschetz Operator and Contraction with the Fundamental Form

Let $M$ be a Kahler manifold, with metric $g$, Kahler form $\omega$, and dual Lefschetz operator $\Lambda$. $\Lambda$ and contraction with $\omega$ both map two forms to 0-forms: are they equal?
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number of eigenvalues = dimension of eigenspace

Is this true in general? What about: number of negative eigenvalues = dimension of span(eigenectors for the negative eigenvalues)? Or even more generally: number of eigenvalues greater than 4.3 = ...
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1answer
23 views

Show that $[A,\exp(B)]=\exp(B)[A,B]$

Denote $\exp(A)=\sum_{k=0}^{+\infty} \frac{A^n}{n!}$ where $A\in M_n(\mathbb{R})$ and $[A,B]=AB-BA$ Assume that $A,B$ commute with $[A,B]=AB-BA$ Show that $$[A,\exp(B)]=\exp(B)[A,B]$$ ...
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19 views

If $\|U(x)\| = \|x\|$, and $x$ is in an orthonormal basis, must $U$ be unitary?

I had this question on my final yesterday. I still don't know the answer. Can someone please tell me the answer/explain it to me? Thanks. Ps: I wrote that yes it must be unitary.
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1answer
15 views

Show linear operator $L$ has no eigenvalue, i.e. there exist no $\lambda \in \mathbb R$ and $f \in C(\mathbb R,\mathbb R)$ s.t. $L(f) = \lambda f$.

Let $C(\mathbb R,\mathbb R)$ denote the real vectorspace of continuous real functions on $\mathbb R$. Let $L: C(\mathbb R,\mathbb R) \rightarrow C(\mathbb R,\mathbb R)$ denote the function $L(f)(x) = ...
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1answer
20 views

Calculating the eigenvalues of this matrix

I have the following matrix asociated to a $f:R⁴\rightarrow R⁴$ endomorphism: $\left( \begin{array}{cccc} 1 & b & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & a & 1 \\ 0 & ...
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1answer
31 views

A basis of a subspace is subset of a basis of the whole space

If $X$ is a vector space with a basis $B$ and $A$ is a subspace of $X$. Does $A $always has a basis subset of $B$? If yes, how should I prove this? If no, we should give an example of a vector space ...
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35 views

Matrices with Operator Norm $\leq 1$

This is a follow-up conjecture to a question I posed yesterday. The proof in that question should extend to show that if a matrix $A=(a_{ij})\in M_n(\mathbb{C})$ has operator norm $\|A\|\leq 1$, then ...
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2answers
16 views

Linear Algebra, geometric multiplicity

I have a matrix and the question says I that I have an eigenvalue of 0. The question asks me to find the geometric multiplicity of that eigenvalue. I know the answer is 4. I just don't ...
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1answer
34 views

Is this set a subspace of $\mathbb{R}^4$?

is this a subspace of R4? 1st criterion is fulfilled, because (85,-58,11,0) is element of L5 But I dont know how to proof the 2nd and the 3rd 2nd says λ element of R a element of L5 a*λ element ...
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19 views

Exercise regarding normal matrices and their spectrum

I hope could get a few hints to this exercise Let $T\in M_n (\mathbb{C})$ be a normal matrix. Let $\lambda \in \sigma(T)$, where $\sigma(T)$ is the spectrum of $T$. Argue that $1_{\{\lambda\}}\in ...
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50 views

Strategy of a purely algebraic proof of Cayley-Hamilton Theorem

Let $p(\lambda)=det(A-\lambda l)$ be the characteristic polynomial of a $n \times n$ matrix $A$. Then $p(A)=O.$ Let $p(\lambda)=p_{0}+p_{1}\lambda+\ldots+p_{n-1}\lambda^{n-1}+p_{n}\lambda^{n}$. ...
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24 views

Is this set a subspace of R2

Is the set L={(8,-3)} a subspace of R2? the first criterion of a subspace is fullfilled: L!=0, because (8,-3) is element of L the second criterion of a subspace is fullfilled too: be there t ...
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43 views

Two locally linearly dependent operators are linearly dependent

Let $S$ and $T$ be operators on a complex vector space $X$. Suppose they are locally linearly dependent, i.e., $Tx=\alpha_{x}Sx \quad \forall x\in X, \quad \alpha_{x}\in \mathbb{C}$. Then we must show ...
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relation between singular values and eigenvalue

How is this inequality proved $\sigma_{min}(A) \leq \min_{i}|\lambda_i|\leq\max_{i}|\lambda_i| \leq \sigma_{max}(A) $ where $\sigma$ are the singular values and $\lambda $ are the eigen values of ...
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Can someone please explain me in details the proof of the following theorem? Help

Let $(V,+,\cdot)$ be a vector space over a field $F$ and let $G$ be spanning for $V$ of size $n$. Let $L$ be a set of linearly independent vectors of $V$ containing $m$ vectors. Then $m\leq n$ and ...
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If a matrix $M$ is positive definite, how can I show that there exists a self adjoint matrix s.t. $M=SS^{T}$

If a matrix $M$ is positive definite, I would like to show that there exists a self adjoint matrix s.t. $M=SS^{T}$. I have a proof, and it comes from operators in Linear Algebra Done Right by Sheldon ...
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45 views

chosing between matrix theory and combinatroics

I have to take one more math course to finish my math minor , i am a computer science major and i want to know which course will benefit me more matrix theory or combinatorics and which takes more ...
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Formula for calculation score based on distance

I try to write function for calculating scoring from distance in my game. I found something similar : link But I need the distance(x) to be between 0-31855000 meters and score between 0 to 1000. ...
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30 views

finite and infinite, vector space, linear transformation

I have to answer these questions for homework and I don't know if I'm answering these correctly. I think most of them are correct, but a double check would be much appreciated. a) If $S$ is a set ...
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Does singular values have anything to do with eigen-values of a square matrix?

We know from linear-algebra, how to calculate the singular values $\sigma_{n(A)}$ of a square-matrix, $A$ by square-rooting the eigen-values of $A^*A$ i-e $\sigma_{n(A)}=\sqrt{\lambda_{n(A^*A)}}$. ...
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29 views

Eigen values and Eigen vectors

Let A be a 4x4 matrix with real entries such that $ \ -1,1,2,-2 \ $ are its eigen values.If $B=A^4-5A^2+5I$ ,where $I$ denotes the 4x4 identity matrix ,then which of the following statements are ...
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3answers
38 views

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 $(R>0, \theta)$ up to angle $\theta+ 2 \pi$.

Prove uniqueness of polar-coordinates $(R>0, \theta)$ 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>0, ...
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1answer
40 views

If a set of 2x2 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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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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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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59 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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70 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 ...
3
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1answer
21 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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1answer
19 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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1answer
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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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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A function $w(t)$ which satisfy $\int dt w(t)F[x](t)=c$ for all x

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 ...