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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186 views

How do I find some distinct vectors u, v, w such that span({u, v}) = span({v, w})?

$\operatorname{Span}(\{u, v\})$ is a set that contains all the linear combinations $au + bv$ where $a$ and $b \in \mathbb R$. $\operatorname{Span}(\{v, w\})$ is a set that contains all the linear ...
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34 views

Find the distance from function $x(t)=t$ to a given subspace of $L^2(-\pi,\pi)$

Let $Z=\operatorname{span}(1,\sin t,\cos t)$, $x(t)=t$. Find $\operatorname{dist}(x,Z)$ in $L_2(-\pi,\pi)$. This question in from my homework. From a lemma that we learned it says if $Z$ is ...
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30 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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61 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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51 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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241 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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100 views

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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27 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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84 views

Dual Lefschetz Operator and Contraction with the Fundamental Form

Let $M$ be a Kahler manifold, with metric $g$, Kahler form $\omega$, Lefschetz operator $L$, and dual Lefschetz operator $\Lambda$. $\Lambda$ and contraction with $\omega$ both map $k$-forms to ...
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158 views

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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96 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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26 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
84 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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27 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
180 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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1answer
43 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
39 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
45 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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25 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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270 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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80 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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46 views

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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60 views

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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61 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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47 views

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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95 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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76 views

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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128 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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156 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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1answer
36 views

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
993 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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55 views

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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121 views

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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69 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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3answers
78 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 ...
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1answer
34 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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51 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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1answer
37 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
33 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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239 views

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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4answers
190 views

Find the axis of rotation of a rotation matrix by $INSPECTION$ (NOT by solving $Kv=v$)

$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$ Find the axis of rotation for the rotation matrix $K$ by INSPECTION. This is from my other thread ...
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311 views

Prove that if $v_1,v_2,…,v_r$ form a linearly independent set of vectors in $V$…

Let $S$ be a basis for an n-dimensional vector space $V$. Prove that if $v_1,v_2,...,v_r$ form a linearly independent set of vectors in $V$, then the coordinate vectors $(v_1)_S,(v_2)_S,...,(v_r)_S ...
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1answer
24 views

Linear transforms question

Let $T_s$ be the counter-clockwise rotation about the positive y-axis through an angle $\varphi$. Write the standard matrix of as $T_s$. I'm not entirely comfortable when questions present ...
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101 views

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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23 views

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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1answer
21 views

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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25 views

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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1answer
56 views

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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1answer
23 views

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 + ...