Tagged Questions

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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9 views

underdetermined homogeneous system of linear equations has always infinitely many solutions

I want to prove that an underdetermined homogeneous system of linear equations always has infinitely many solutions. I know that an homogenous system of linear equations always has the trivial ...
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1answer
15 views

How are eigenvalues relevant to the invariants of a system?

For a matrix $\mathbf{A} \in \mathbb{R}^{2\times2}$, what can one say about its eigenvalues $\gamma_1, \gamma_2 \in \mathbb{C}$, if: $$\mathbf{S}_0 \in \mathbb{R}^{2}$$ $$\mathbf{S}_{n+1} = ...
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1answer
27 views

Question on Norm and Inner Product

Polarisation identity states that $\langle x, y\rangle = \frac{1}{4}\|x+y\|^2 - \frac{1}{4} \| x - y \|^2$. And this is proven by expanding the terms on the right using $\|x\|^2 = \langle x,x\rangle ...
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2answers
32 views

Suppose $KA = {\bf0}$ and $K$ is idempotent. Define $G = (A-K)^{-1}$. Prove that (i) $AG = I-K$; (ii) $AGA = A$; and (iii) $AGK = {\bf0}$.

I don't know how to start this one. Should I divide these into cases where $K$ is the identity matrix, the null matrix and an idempotent matrix w/c is not null and identity? Help please. Thank you.
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21 views

Interpreting results from a linear system

$$\begin{pmatrix}t&1&1\\1&t&1\\1&1&t\end{pmatrix}\cdot\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=\begin{pmatrix}1\\1\\1\end{pmatrix}$$ Is a linear system. Find the values of t ...
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2answers
27 views

Is the set of matrices with rank at most $r$ closed? [duplicate]

The question is as follows: $\DeclareMathOperator{\rank}{rank}$ Is the set $S_r = \{A \in \Bbb R^{n \times n}: \rank(A) \leq r\}$ closed in $\Bbb R^{n \times n}$ in the Euclidean topology? I ...
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0answers
4 views

Finding Transformation Matrix from source/destination vector pairs dataset

I have a color processing problem I'm trying to solve. When we convert RGB colors to different colorspaces, we use 3x3 matrices. For example if s is a source RGB color vector, M is a 3x3 colorspace ...
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19 views

Trying to solve this system with Gauss-Seidel

I'm trying to solve this system: $$ \begin{cases} {-x}+5y+3z=2\\ 7x+4y+2z=7\\ 3x-y+5z=5 \end{cases} $$ I have to use Gauss-Seidel, but no matter how I try the system does not converge. So my question ...
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0answers
12 views

Calculate area defined by matrix equation

Suppose I have an $n$-dimensional matrix $A$. I define a region as being the set of all vectors ${\bf x}$ such that when I calculate $A.{\bf x}$ the resulting coordinates are all between 0 and 1. (or ...
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19 views

Find points in a reference unit square

First of all sorry if this question has been answered or is in the wrong place. As part of an algorithm, I need to map two points $(P_0,P_1)$ in an arbitrary quadrilateral to a reference unit square ...
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1answer
16 views

Show that S is a real vector space using the standard operations on R3 (what are “standard operations” on R3?)

My problem is: Let S = {(x,y,0):x,y E R}. Show that S is a real vector space using the standard operations on R3. what exactly are the standard operations on R3? I'm not sure if it means closed ...
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0answers
8 views

Averaging and approximation

I read a paper reference at http://arxiv.org/pdf/1101.1764.pdf that if we average a set $V=\{V(t_0,\nu_0), V({t_1,\nu_1),..., V(t_n,\nu_n)}\}$; with $V(t_i,\nu_i)=e^{i\sigma(t_i,\nu_i)}$ then we can ...
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1answer
28 views

Show that the $\|f\|_2\le\|f\|_\infty$ in $X=C[0,1] $ [on hold]

In $X=C[0,1] $, show that $\|f\|_2 \le \|f\|_\infty$.
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2answers
19 views

Plane passing through 2 points, parallel to a vector

So I am given two points P and Q and a vector $\vec v$ in $R^3$. I want to find the plane that goes through P and Q and is parallel to v. So I made a vector PQ, but how can I make that into plane ...
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0answers
6 views

Relations involving image and kernel of endomorphisms

Let $f,g$ be endomorphisms of a $K$-vectorspace $V$, $V$ being finite-dimensional. We are given that $\text{im}(fg)=\text{im}(gf)$ and that $\text{im}\,f+\text{ker}\,g = V$. Prove that ...
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2answers
30 views

Determinants of 'block' matrices

I am trying to simplify the determinant of \begin{pmatrix}C&A\\B&0\end{pmatrix} where $A$ and $B$ are square $m\times m$ and $n\times n$ matrices, and $C$ is some $m\times n$ matrix, $0$ is ...
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1answer
18 views

Is the set a basis for the Subspace

Given the subspace $W= \{(x_{1}, x_{2}, x_{3}): x_{1} + x_{2} + x_{3} = 0\}$ Is the Set $S= \{(-1, -1, 2), (-3, 2, 1)\}$ a basis for W. What I did was, I first checked if it was linearly independent ...
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1answer
15 views

question regarding the geometric meaning of eigenvalues and eigenvectors

Ok so I've known how to get eigenvalues and eigenvectors for a while, but am becoming more interested in a 'simple' explanation of what actually is going on. I've looked up things on Google etc. ...
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0answers
11 views

General Steinitz exchange lemma

Where can I find a proof of the following general Steinitz exchange lemma: Let $B$ a basis of a vector space $V$, $L\subset V$ be linearly indepdent. Then there is an injection $j:L\rightarrow B$ ...
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4answers
58 views

Strange solution after dividing equation

I have $$3x=0$$ equation. I divided both sides of it by x and got: $$\frac{3x}{x} =\frac{0}{x}$$ $$3 = 0$$ I want to ask, how is that possible? What did I do wrong? Did I break any rule of math?
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1answer
24 views

matrix derivative w.r.t a scalar

How the following derivative can be calculated? $\displaystyle\frac{d}{d\lambda}A\left(\lambda\ I+A^TA\right)^{-1}A^T$ where $A$ is a rectangular matrix and $\lambda$ is a scalar.
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0answers
18 views

A Simple Bound on Super-Additive Functions

If $f(x)$ is a positive super-additive function ($\sum f(x) \leq f(\sum(x) $), can we prove that: $$I = \sum_i f\left(\sum_j x_{ij}\right) + \sum_j f\left(\sum_i x_{ij}\right) - 2 \sum_i \sum_j ...
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1answer
22 views

upper bound on a matrix norm

what is the smallest upper bound for the following norm $\|\left(\lambda\ I +A\ A^T\right)^{-1}\|<?$. where, A is a rectangular matrix, $\lambda>0$ is a scalar. (any possible norm)
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1answer
33 views

what does “closed subspace” in papers mean?

In many books and articles one finds sentences like this: "let $A$ be a closed subspace of ...". Now my question might be stupid, but I am always wondering what they mean by closed subspace? Is this ...
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2answers
20 views

Showing a set is a subspace

Let $X$ denote the set of function $[0,1] \to \mathbb{R} $ and $X$ is a real linear space. Define $A$ and $B$ by: $A = \{x \in X \mid x(0) = 0\}$ $B = \{y \in X \mid y(1) = 0 \}$ Show $A$ and $B$ ...
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1answer
14 views

System of linear equations, 2 solutions

I'm thinking about an easy proof, why an System of linear equations can't have 2 solutions. I know that it can only have 0, 1 or infinitely many. But why are only these possibilities possible?
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0answers
25 views

Recovering flow values given total values

I have the following problem which I am failing to put into a tractable Mathematical minimization problem. We are observing some flows. A flow can start at any month in a year and end in any month ...
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0answers
11 views

Give an example of a matrix reduce to the cononcial form. Also find the non singular matrix P and Q such that PAQ is in the cononical form. [on hold]

Give an example of a matrix reduce to the cononcial form(normal form). Also find the non singular matrix P and Q such that PAQ is in the cononical form(normal form).
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0answers
22 views

Showing thislinear operator on an inner product space is its own transpose

Let $H$ be the inner product space of continuous real valued functions defined on $[0,1]$ where $(\alpha\mid\beta)=\int_{0}^{1} \alpha(u)\beta(u)du$ Put $K(s,t)=\min\{s,t\}-st$. Define $T∈L(V,V)$ by ...
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1answer
14 views

Show tha the yz-plane is spanned by thes vector [on hold]

Show that yz-plane w={(0,y,z):ybelongs to R} is spanned by (0,1,1) and (0,2,-1)
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0answers
7 views

Linear Transformations adn linear functionals

$F$=any field of characteristic 0. $V$=$F^3$, $W$=$F^4$ p∈L(V,W) given by p((x,y,z))=3x+4y+2z; q∈L(W,F) given by q((w,x,y,z))=2w+5x+7y+11z; T∈L(V,W) given by T((x,y,z))=(x,x+y,x+y+z,y+z) ...
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0answers
28 views

Show that $deg(f\cdot g)=n+m$

I started learning about rings and I was asked to proof some claims. I don't understand how I may prove the last one. I have proven that if $f$ and $g$ are polynomials over some ring of polynomials, ...
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0answers
27 views

$x \cdot x$ in inner product space is a quadratic form

Given an inner product space with some inner product $\cdot$ , how can I prove that $x \cdot x$ for any vector $x= (x_1,... x_n)$ is a quadratic form in $x_i$? I know how to recover an inner product ...
3
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0answers
23 views

Range of vectors that turn into eigenvectors after recursive multiplication by a matrix

Suppose $\mathbf{x}$ is a vector, and $\mathbf{A}$ is a square matrix. Which $\mathbf{x}$'s will satisfy the equation $\mathbf{A}^n\mathbf{x} = \lambda\mathbf{A}^{n-1}\mathbf{x}$, where $\lambda$ is ...
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0answers
24 views

problem in linear algebra [on hold]

Prove: If $A$ is invertible, then $AB^{-1}$ and $1+BA^{-1}$ are both invertible OR both not invertible
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2answers
33 views

Hilbert space: product and tensor product space

Let $H_1$ and $H_2$ be Hilbert spaces, then I would intuitively define the inner product on $H_1 \times H_2$ by $\langle (x_1,x_2),(y_1,y_2)\rangle = \langle x_1,y_1 \rangle + \langle x_2,y_2 ...
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2answers
45 views

Definition of sign

The following definition is in my notes with no explanation: $$\operatorname{sgn}(\sigma)=\begin{cases}1,&\text{if }\sigma(p)(x_1,\ldots,x_n)=p(x_1,\ldots,x_n)\\-1,&\text{if ...
2
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1answer
44 views

Develop good understanding of Linear Algebra

I am self studying Linear Algebra from book by Kenneth Hoffman and Ray Kunze, and currently I'm on 2nd Chapter of vector spaces, though the text is easy to follow, specially the exercises that follow ...
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0answers
24 views

Approximating Averaging : Signal processing

I read a paper reference at http://arxiv.org/pdf/1101.1764.pdf that if we average a set $V=\{V(t_0,\nu_0), V({t_1,\nu_1),..., V(t_n,\nu_n)}\}$; with $V(t_i,\nu_i)=e^{i\sigma(t_i,\nu_i)}$ then we can ...
0
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0answers
13 views

Is it possible to have all the rows distinct(unique) in a matrix having only 0 or 1 , after removing exactly one column?

I have to solve the following programming problem Given a M*N binary matrix. Detect if it is possible to delete a column in a manner that after deleting that column, the rows of the matrix will be ...
0
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2answers
21 views

Find the matrix $A$ with this condition…

If $\theta \in\mathbb{R}\setminus\{k\pi, k\in\mathbb{Z}\}$ and $A\in M_{2\times 2}(\mathbb{C})$ such that $$A^{-1} \begin{pmatrix} \cos \theta & -\sin\theta \\ \sin \theta & ...
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1answer
21 views

Equivalent of solutions of IVP

Consider the IVP $y''-2y'+26y=0$, $y(0)=1$, $y'(0)=2$. From the characteristic equation $m^2-2m+26=0$, i found the roots as $m_1=1-5i$ and $m_2=1+5i$. Then when i use the basis solutions ...
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0answers
31 views

One question on Matrix Equation

Assume $\hat{M}_1, \hat{M}_2, \hat{T}_{11}, \hat{T}_{12}, \hat{T}_{21}, \hat{T}_{22}$ are $2\times 2$ matrix. And $a, b, A, B, C, D$ are all numbers, satisfying the following relation: \begin{align} ...
2
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0answers
20 views

Is there a nice representation for KKT conditions for matrix constraints?

I have a convex programming problem: $\min \left\lVert J - R \right\rVert _2$ $J,R$ are matrices. $J$ is given for the problem. One of the constraints is: $R = KQ$ Here, $R,K,Q$ are matrices. $K$ ...
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1answer
26 views

If A is a Hermitian matrix then SAS* is Hermitian

If $A$ is an $n\times n$ Hermitian matrix, and $S$ is an nxn matrix, then $SAS^*$ is also Hermitian. Why is this true? I have seen this claim made in several places but can't find a proof.
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2answers
33 views

Are three vector not in one plane mutually orthogonal, or linearly independent? [on hold]

Let $u, v, w$ be three points in $R^{3}$ not lying in any plane containing the origin. Are these three points linearly independent or mutually orthogonal?
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0answers
14 views

Can the Lanczos algorithm converge very fast by choosing initial guess smartly?

Suppose I have the two lowest eigenvectors $v_1$, $v_2$ of a matrix $M$. If slightly change $M$ to $M'$. Can I use $v_1$ or $v_2$ as an initial guess for $M'$? If so, which one should be used, $v_1$ ...
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0answers
30 views

Prove $NuclearNorm(W*U*S)\geq NuclearNorm(W*S)$

Suppose $W$, $S$ is two diagonal matrices of size $n*n$. $U$ is an orthogonal matrix. For $W$, the diagonal elements satisfies: $0\leq w_{1,1}\leq w_{2,2}\leq ...\leq w_{n,n}$, and for $S$, the ...
1
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1answer
20 views

Find a basis of a subset given an equation

$W = \{(x_{1}, x_{2}, x_{3})\in $R$^3: \frac{x_{1}}{3} = \frac{x_{2}}{4} = \frac{x_{3}}{2}\}$ Find a basis for $W$ I need help. I don't know how to do this.
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0answers
26 views

Prove $NuclearNorm(W*U*S)\geq NuclearNorm(W*S)$

Suppose $W$, $S$ is two diagonal matrices of size $n*n$. $U$ is an orthogonal matrix. For $W$, the diagonal elements satisfies: $0\leq w_{1,1}\leq w_{2,2}\leq ...\leq w_{n,n}$, and for $S$, the ...