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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Uniqueness in Matrix Multiplication

I'm sure there is an answer to this somewhere else, but I'm simply not sure how to find it or what to call it. I looked online, but couldn't find anything. The question is as follows: Let $A$ and ...
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2answers
36 views

Why does this form a basis for $V$? (Intuitive explanations please)

Let $V$ be the space spanned by $\mathbf f_1=\sin(x)$ and $\mathbf f_2=\cos(x)$. Show that $\mathbf g_1=2\sin(x)+\cos(x$) and $\mathbf g_2=3\cos(x)$ form a basis for $V$. We can see that $$\mathbf ...
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1answer
11 views

Help trying to identify a set and determine whether it is a subspace of $\Bbb{R}^n (n>2)$

I'm trying to figure out what this set is $\{x \mid \sum_{j=1}^{n}x_j =0\}$. Also any hints on how to show this is a subspace of $\Bbb{R}^n (n>2)$?
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1answer
27 views

On representation of quadratic form

In linear algebra, a quadratic form is defined as $Q(x)=x^TAx$ for some (non-singular) matrix $A$ and any $x\in V$, where $V$ is a vector space. Actually, quadratic form can be any one satisfying ...
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2answers
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Prove that the vectors $v_1,v_2,\ldots,v_k \operatorname{span}R^n$ if and only if $[v_1]_B,[v_2]_B,\ldots,[v_k]_B \operatorname{span}R^n$.

From section on Change of Basis $\longrightarrow$ Assume the vectors $v_1,v_2,\ldots,v_k\operatorname{span}R^n$, we must show that $[v_1]_B,[v_2]_B,\ldots,[v_k]_B\operatorname{span}R^n$. We can ...
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1answer
27 views

Determine which of the following subsets of $\Bbb{R}^n$ are subspaces of $\Bbb{R}^n (n>2)$.

I'm having a bit of trouble showing that the following subsets of $\Bbb{R}^n$ are subspaces of $\Bbb{R}^n (n>2)$. I know that I need to show that they are closed under addition and multiplication, ...
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1answer
21 views

linear algebra on cramer's rule

Verify the following system of linear equation in $\cos{A}$,$\cos{B}$ and $\cos{C}$ for the following triangle equation: $c\cos{B} + b\cos{C} = a$, $c\cos{A} + a\cos{C} = b$ and $b\cos{A} + a\cos{B} = ...
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6answers
117 views

Construct an endomorphism $f$ such that $f\circ f=-Id$

Let $E$ be a real vector space of finite dimension $n$ and $f$ an endomorphism such that $$f\circ f=-Id_E$$ Show that $n = \dim (E)$ is an even integer Assume $n$ is even, $n=2p$. ...
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1answer
19 views

Direct product norm

Given a norm on $V$ say $||*||$, what is the norm on $V \times V$? Can we induce this norm from $||*||$? Please help with understanding this.
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1answer
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Equation of hyperplane in Matlab

Given $n$ points in $n$-dimensions, using MatLab, how should we find the equation of the $(n-1)$-dimensional hyperplane passing through these $n$ points.
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5answers
81 views

Book recommendation for Linear algebra.

I am looking for suggestions, it has to be a self study book and should be able to relate to applications to real world problems. If it is more computer science oriented , that would be great.
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2answers
25 views

Is there a way to “transpose” from scalar product EA.EB to AE.BE?

I am wondering if there is an easy way to "transpose" from the result of the scalar product EA.EB to AE.BE ? In my case, EA.EB = 1/2
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0answers
24 views

Least squares polynomial approximation $(f-p_n,q)=0$ proof.

I know how to do the other way around but I am getting stuck with showing the following If $<f-p_n,q>=0$ then $p_n$ is a polynomial of best least squares approximation in a norm $|\cdot|$ for a ...
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0answers
94 views
+50

Set geometry and inclusion

I would like to prove that the set of the symmetric positive semi-definite matrices which is defined as $\Delta_2= \{S\in\mathbb{S}_{m,m} \quad \text{s.t.}\quad ...
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1answer
20 views

Inner products and distributive property

Is this true for the inner products ? : $(\vec a + \vec b)\cdot(\vec c + \vec d) = \vec a\cdot\vec c + \vec a\cdot\vec d + \vec b\cdot\vec c + \vec b\cdot\vec d$.
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0answers
19 views

Basic Question Linear Transformation and Matrix computations

Can someone show me how to do this question? http://imgur.com/cIciHnY I'm studying for a test and this was a question off a past test. I would love to show my thoughts but I do not know how to format ...
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1answer
136 views

Existence of a basis such that $\|e_i\|=1$ and $\|e_i^{*}\|_*=1$. (dual)

Let $E$ a $n$-finite dimensional normed vector space. Can we find a basis $e_1,e_2,\cdots,e_n$ of $E$ such that $\|e_i\|=1$ and $\|e_i^{*}\|_*=1$ for all $i$ ? where $\|\|_*$ is the dual ...
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0answers
18 views

QR Algorithm with Shifts Question

Why must QR Algorithm with Shifts make no progress when applied to this n x n matrix? (attached as image). Also, if a matrix A is orthogonal in a QR factorization, will R be tridiagonal? How would ...
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2answers
22 views

Show that the set $W$ of all polynomials in $P_2$ such that $p(1)=0$ is a subspace of $P_2$. Find a basis for $W$.

a.) Show that the set $W$ of all polynomials in $P_2$ such that $p(1)=0$ is a subspace of $P_2$. b.) Make a conjecture about the dimension of $W$. c.) Confirm your conjecture by finding a ...
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1answer
8 views

Algorithm to find the lower envelope of set of piece-wise linear functions

I am looking for an algorithm that finds the lower envelope of a set of continuous piece-wise linear functions. E.g. given two functions $f(x)$ and $g(x)$ I want to find $h(x)$ as shown below: ...
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1answer
36 views

Change of basis?

So the question is... A transformation $T$ is denoted by $T(x,y)=(x+y,x-y)$. $C$ is the basis $\{(1,-1),(1,1)\}$ $D$ is the basis $\{(1,2),(1,0)\}$ I know $T(C)=\{(0,2),(2,0)\}$ But how do I ...
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1answer
35 views

Linear maps and matrix coefficients

I am currently working through this page in my script: Can somebody explain what this means and how it works in practice? Perhaps if I saw an example I could follow it. Thanks for your help!
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0answers
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Linear algebra.Proof proportinal between minors and cofactors

$B$ is square matrix. Order of matrix $B$ is $n$. First $m$ lines form the matrix $C$, $rank (C)=m$.Last $n-m$ lines form fundamental system solutions of homogeneous linear equation with matrix $C$ ...
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1answer
24 views

Collinear points in 3dimension

Given three $3D$ points: $A,B$ and $C$, what is the procedure to check if they are collinear? In general, given $n$ points in $m$-dimension, how should one find out, if these $n$-points defines a ...
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0answers
15 views

Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
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0answers
12 views

Books in spectral theory for finite dimensional spaces

I'm looking for beginner books of spectral theory for finite dimensional spaces. I've already heard about this subject, but I don't know where I can find it. What's the domain of this subject? (Linear ...
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0answers
22 views

Show $W_1 \hookrightarrow V \twoheadrightarrow W_2$ is an isomorphism

Let $\langle , \rangle$ be a non-degenerate bilinear form with the signature $(p,q)$ on a real vectorspace $V$ and $W_1, W_2$ subspaces, such that the restriction $\langle , \rangle |_{W_i}$ is ...
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1answer
13 views

Using Givens Rotation on a vector

Say we have a vector v=$[3\ 0\ 4]$. Find a 3x3 orthogonal matrix Q such that only the second component of Qv is nonzero and such that this component is also positive. Is Q unique? I tried ...
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0answers
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Can I modify a polynomial to return only multiples of a given number?

I'm attempting to create a polynomial equation for a project of mine, with a shape similar to the following: $${3x^5\over500}+{x^4\over25}+x^3+40 x^2+100 x$$ However, one of my goals is to have the ...
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0answers
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Confused about the association between two vectors

Polynomial $x^3 + 2x^2 + 4 \in P_3(\mathbb R)$ and $(1, 2, 0, 4) \in \mathbb R^4$. $x^3 + 2x^2 + 4$ is equivalent to $(1, 2, 0, 4)$ apparently because $(1, 2, 0, 4)$ $= 1 (1, 0, 0, 0) + 2 (0, 1, ...
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4answers
2k views

Why does cross product give a vector which is perpendicular to a plane

I was wondering if anyone could give me the intuition behind the cross product of two vectors $\textbf{a}$ and $\textbf{b}$. Why does their cross product $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
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1answer
495 views

Least square solution based on the pseudoinverse solved efficiently with singular value decomposition

Hi apologies it's hard to type out the problem, I have a lecture slide on neural networks. It says the fitting error gives the matrix: N by M matrix of thi's multiplied by Mx1 weights minus Nx1 ...
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0answers
15 views

Inner Product inequality problem using Cauchy Schwarz, or what other way?

Let $<p,q>$ be an inner product on n. If p and q are both of degree n, show that $<p,q>^2$ $\leq$ $<p,p>$ $<q,q>$. I tried multiplying the right side out but am getting ...
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1answer
18 views

Multicollinearity and SVD

I compute the Singular Value Decomposition of a n x n matrix. If the matrix is not full rank, and I have 2 collinear columns, I end up with one singular value equal to 0. Is it possible to find out ...
3
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1answer
33 views

Show: $\varphi\colon\mathbb{Z}_{mn}\to\mathbb{Z}_m\times\mathbb{Z}_n, k\mapsto (k\% m,k\% n)$ is a ring isomorphism for $m$ and $n$ relatively prim

Let $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Let $m,n\in\mathbb{N}$ be relatively ...
5
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1answer
391 views

Anti-commutative matrices

If $A$ and $B$ are anti-commutative square matrices, so $AB+BA=0$, how do you a) prove that $\mathrm{tr}(A)=\mathrm{tr}(B)=0$ and b) prove that the order of the matrices is even?
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1answer
279 views

Spectral radius and positive definite of matrices

Denote $ \rho(A)$ to be the spectral radius of a matrix $A,$ that is the maximal eigenvalue of $A.$ We say that a matrix $M$ is positive definite, respectively positive semidefinite, if $x^TMx>0$ ...
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2answers
59 views

Dual Vector Space embedding

Is there an embedding of any vector space $V$ into $V^*$? As far as I know it is not true. The statement that I know of is that there is natural embedding of $V$ into $V^{**}$ Is there any ...
5
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4answers
209 views

if $AB\neq 0$ for any non zero matrix $B$ then $A$ is invertible

Question is to check that : If $A$ is an $n\times n$ matrix over a field $F$ and $AB\neq 0$ for any non zero matrix $B_{n\times n}$ over $F$ then, $A$ is invertible. This does make some sense to me ...
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1answer
32 views

Proof of Eckart-Young-Mirsky theorem

Could someone please explain why in http://en.wikipedia.org/wiki/Low-rank_approximation#Proof_of_Eckart.E2.80.93Young.E2.80.93Mirsky_theorem it says "we know that $\exists(k+1)$ dimension space ...
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1answer
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Compute the vector $v$ if the coordinate vector $[v]_{s}$ is given with respect to each ordered basis $S$ for $V$

Ok, so this is a practice question in my book: $V$ is $M_{22}$ $S=$ \begin{bmatrix} 1&-2\\ 0&0\\ \end{bmatrix} \begin{bmatrix} -1&3\\ 0&1\\ \end{bmatrix} \begin{bmatrix} 1&0\\ ...
3
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0answers
35 views

If $A \in M_{n,n}(\mathbb F)$ is invertible then $A = UPB$, $U$ is unipotent upper triangular, $B$ is upper triangular and $P$ is a permutation.

If $A \in M_{n,n}(\mathbb F)$ is invertible then $A = UPB$, where $U$ is unipotent upper triangular, $B$ is upper triangular and $P$ a permutation matrix. A hint is given that one could relate ...
3
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1answer
210 views

Column and Row Picture for Singular System of 100 Equations (Strang P55, 2.2.32)

Start with 100 equations $\color{#8F00FF}{A}\mathbf{x} = \mathbf{0}$ for $\mathbf{x} = (x_1, ..., x_{1oo})$. Suppose elimination reduces the 100th equation to $0 = 0$, so the system is "singular". ...
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1answer
23 views

A is a Hermitian projection if and only if it is an orthogonal projection

I need to figure out this property of Hermitian / Orthogonal projections "A is a Hermitian projection if and only if it is an orthogonal projection" Your assistance will be highly appreciated. ...
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1answer
78 views

Valid Proof for Cayley Hamilton Theorem? (Not the usual incorrect one)

By induction; case n=1 is true. $A$ admits an eigenvalue $\lambda$ with eigenvector $v$ over $\mathbb{C}$. Change $A$ into a basis $e_1=v,...,e_n$. Then $\exists X$ such that $XAX^{-1}=\left( ...
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3answers
60 views

An infinite generating set of a finite dimensional vector space contains a basis

Let $S$ be an infinite generating set of a finite dimensional vector space , then how do we prove that there is a subset of $S$ which is a basis of the vector space ? Please help
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4answers
28 views

The row rank of an $m\times n$ matrix $A$ is at most $\min\{m,n\}$. Why?

Ok, so let $A$ be an $m\times n$ matrix. I understand by intuition that the row rank has to be $\le m$, but why also $n$? Is this because there can be no more leading ones than $m$ or $n$?
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0answers
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Odd and Even Weight functions in orthogonal polynomials proof

Suppose now that w is an even function, i.e. $w(-x)$ = $w(x)$ for all x in $[-1,1]$ and let $p_0$,..., $p_n$ be a family of orthogonal polynomials with respect to w. Prove by induction that $p_k$ is ...
4
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1answer
42 views

Canonically isomorphic but not equal

In mathematics, we have many objects that are canonically isomorphic but not equal on the nose. For example let $V$ and $W$ be vector spaces. Then $V\otimes W$ and $W\otimes V$ are canonically ...
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4answers
112 views

How to prove a set of vectors does not span a space.

Ok, so I'm a bit curios as to how you can prove a set does not span a vector space. For example, let ${S}$ be the vector set \begin{bmatrix} 1\\ 0\\ 0\\ 0\\ \end{bmatrix} \begin{bmatrix} 0\\ 1\\ 0\\ ...