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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2
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2answers
15 views

Jacobian matrix for $f(h)=hh^Th$, where $h$ is an $m$ dimensional vector

I have a function $f(h)=hh^Th$, can we say $\nabla f(h)=2*hh^T + h^ThI_{m\times m}$, where $I_{m\times m}$ is an identity matrix?
0
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1answer
29 views

question about span and basis

I have a question from homework (I'm not sure if my solution is correct): Let $V$ be a vector space over a field $\mathbb{F}$ and let $W$ be subspace of $V$. Let $u$ and $v$ be vectors in ...
0
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1answer
53 views

How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
1
vote
1answer
33 views

Suppose $\mu$ is not an eigenvalue of A. Show that the equation $x'= Ax + e^{\mu t}b$.

Suppose $\mu$ is not an eigenvalue of $A$. Show that the equation $x'= Ax + e^{\mu t}b$ has a solution of the form $\varphi(t) = ve^{\mu t}$.
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2answers
24 views

Shown that this matrix is a representative of a group.

I have to show that this matrix (for which $ad - bc = 1$) \begin{pmatrix} a & b & 0\\ c & d & 0 \\ 0&0&1 \end{pmatrix} is a representative of a group. The same for this one: ...
0
votes
1answer
49 views

Solve this problem?

Here I have 2 lists, A and B. I am trying to find the connection items of list A got with items of list B. I know: b) all the items of A a) the 1st item of B b) number range from 0-255 A - B 0 ...
0
votes
1answer
17 views

proving determinant of lower triangular matrix from definition

we define $$det(A) = \sum_{\sigma \in S_n} (sgn\sigma)a_{1\sigma(1)}...a_{n\sigma(n)}$$ Matrix $A = (a_{ij}) $ where $a_{ij} = 0$ for $j>i$ and I want to use it to prove the determinant of a lower ...
3
votes
1answer
89 views

counterexamples to $ \det \Big(A^2+B^2\Big)\ge \det(AB-BA) $

$n\geq3$. A and B are two $n\times n$ reals matrices. For $n\times n$, Could one give counterexamples to show that $$ \det \Big(A^2+B^2\Big)\ge \det(AB-BA) \tag{$*$}$$ is not necessarily true? ...
1
vote
3answers
35 views

Standard matrix A of T?

Help please. What would be the standard matrix of A? I know how to do number 2 and 3 but I'm just having trouble with A. I asked this earlier but I lost my account and I'm not sure if I posted ...
0
votes
2answers
22 views

Finding eigenvalues of indefinite matrix

Let V = M_2x2(R). Let T: V to V be defined by T(a,b,c,d) = (d,b,c,a). Find the eigenvalues of T and a basis B of V so that [T]_B is a diagonal matrix. I find the only eigenvalue to be 0. In this ...
0
votes
3answers
30 views

For two p.d. matrices $A$ and $B$, prove that $\lambda_1(AB)\leqslant \lambda_1(A) \cdot\lambda_1(B)$

If $A$ and $B$ are two nxn positive definite matrices, then show that $$\lambda_1(AB) \leqslant \lambda_1(A) \cdot \lambda_1(B),$$ where $\lambda_1(\cdot)$ denotes the largest eigenvalue.
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1answer
19 views

Let A : $R^2$ to $ R^2$ be a linear transformation with eigenvalues 2/3 and 9/5

Let A : $R^2$ to $ R^2$ be a linear transformation with eigenvalues 2/3 and 9/5 . Then, there exists a non-zero vector $v$ in $R^2$ such that (a) $||Av||$ > 2$||v||$; (b) $||Av||$ < 1/2$||v||$; ...
0
votes
3answers
38 views

Find the rank of the given matrix

Let $x_1$,$x_2$,$x_3$,$x_4$,$y_1$,$y_2$,$y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a 4 x 4 matrix A by A = $$\begin{pmatrix} x_1^2 + y_1^2 & x_1x_2 + y_1y_2 ...
1
vote
1answer
30 views

Continuity of bilinear maps

Given a vector space $V$ over $\mathbb{R}$ with a norm $||*|| $. Can $(x,y)\rightarrow(x+y)$ be an example of continous bilinear map, if yes, can you please exlain why? Definition of continuous ...
0
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1answer
19 views

Metric induced from norm

I was trying to understand the following: Every norm on $R^n$ is continuous (as a map from $R^n$ to $R$). Proof. We use the maximum metric on $R^n$: $ d(x, y) = \max{|x_j − y_j| : j ∈ \{1, . . . ...
0
votes
1answer
13 views

Can this example have been done using row space instead of column space?

Can this example be done using row space instead of column space? I have tried but I am new so don't know if I am doing it correctly, doesn't seem right to me. I tried expressing the given vectors ...
1
vote
1answer
48 views

Prove that this matrix is not diagonalizable WITHOUT determinants

I have this matrix: $ \left( \begin{array}{cccc} 22 & 23 & 10 & -98\\ 12 & 18 & 16 & -38\\ -15 & -19 & -13 & 58 \\ 6 & 7 & 4 & -25 \end{array} \right) ...
0
votes
1answer
28 views

Change of coordinate matrix

Let $T: \mathbb{R}_2 \rightarrow \mathbb{R}_2$ be defined by $T(a,b) = (a+2b, 3a-b)$. Let $B = [(1,1),(1,0)]$ and $C = [(4,7),(4,8)]$. Find $[T]_B$ and $[T]_C$ and show that $[T]_C = Q^{-1} [T]_B Q$ ...
0
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2answers
23 views

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 ...
0
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1answer
17 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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2answers
38 views

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 ...
0
votes
1answer
21 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
28 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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votes
1answer
30 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} = ...
0
votes
1answer
30 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 ...
0
votes
2answers
41 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 ...
1
vote
1answer
22 views

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.
2
votes
5answers
136 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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votes
2answers
27 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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1answer
22 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
24 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 ...
0
votes
0answers
27 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 ...
0
votes
2answers
25 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 ...
0
votes
0answers
21 views

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 $C$ ...
4
votes
1answer
39 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 ...
2
votes
1answer
27 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 ...
0
votes
1answer
9 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: ...
4
votes
1answer
28 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 ...
0
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1answer
39 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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1answer
16 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
20 views

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, ...
0
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0answers
16 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 ...
1
vote
1answer
24 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 ...
2
votes
0answers
33 views

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 ...
0
votes
1answer
16 views

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
votes
1answer
37 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 ...
1
vote
1answer
34 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 ...
1
vote
4answers
29 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 $n$?
3
votes
0answers
38 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 ...
1
vote
4answers
114 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\\ ...