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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0answers
7 views

Why is the dimension of the generalised eigenspace equal to the algebraic multiplicity?

Why is the dimension of the generalised eigenspace equal to the algebraic multiplicity? Can someone provide me with the proof for this? I can't figure out myself.
4
votes
2answers
37 views

Rank-nullity theorem for free $\mathbb Z$-modules

From linear algebra we know that given vector spaces $V$, $W$ over a field $k$ and a linear map $f\colon V\to W$ we have $$\dim V = \dim \operatorname{im} f + \dim \ker f.$$ Is this still true when ...
0
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0answers
16 views

Is the projection of a vector onto a matrix always exist [on hold]

If I set Ax is the projection of vector b in matrix A. Then I get $A^T(Ax-b) = 0$. After ...
0
votes
0answers
4 views

How do I find F(r(t)) for the work done over line integrals?

I don't know what this notations means what does $$ F(r(t)) $$ mean ? Are they asking me to find the force in terms of r(t) ? The force vector just has xs and ys in it ? How is it treated when if the ...
0
votes
0answers
15 views

Find an orthogonal matrix $P$ such that $P^{T}AP$ is diagonal.

I began by finding the eigenvalues and eigenvectors of $A$ where $A=\begin{pmatrix} 4 & 0 & -2 \\ 0 & 2 & -2 \\ -2 & -2 & 3 \end{pmatrix}$. This gave $\lambda_1=0, ...
0
votes
0answers
10 views

Why is the state transition matrix unique if the fundamental matrix is not?

For a linear time variant system, the state transition matrix $\Phi(t,t_0)=X(t)X^{-1}(t_0)$ but you can select any linearly independent initial conditions to build the fundamental matrix $X(t)$, so ...
0
votes
1answer
21 views

Orth0normal in vector space

Suppose that $U$ is a subspace of a vector space $K$, $\{u_1,...,u_k\}$ is an orthonormal basis of $U$, $\{j_1,...,j_m\}$ is an orthonormal basis of $U^\perp $, Prove that $\{u_1,..,u_k,j_1,...,j_m\}$ ...
0
votes
1answer
16 views

What ist the rank of vector system in the following vector space?

I had some difficulties with this quite difficult problem. The vector space is $\Bbb Z_{2}^n$, we have $(1,1,0,\dots,0)$, $(0,1,1,0,\dots,0)$, $\dots$, $(0,\dots,0,1,1)$, $(1,0,\dots,1)$. What is ...
0
votes
2answers
27 views

Matrix Algebra Questions

I would like you to help me with two questions I am stuck in. You can call these homework questions. It would be helpful if you can give me non-trivial hints instead of complete solution. 1) Let $A$ ...
0
votes
1answer
21 views

eigen values and eigen vectors of the projection

Assume that $W$ is n-dimensional subspace of an m-dimensional vector space $V$. Find all eigenvalues and all eigenvectors of the projection operators $P_W$. Here is my ideas: Since $W$ is ...
0
votes
1answer
16 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 ...
1
vote
1answer
18 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} = ...
1
vote
1answer
34 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 ...
0
votes
1answer
36 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.
0
votes
0answers
29 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 ...
2
votes
2answers
30 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 ...
1
vote
0answers
7 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 ...
0
votes
0answers
24 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 ...
1
vote
0answers
16 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 ...
0
votes
0answers
22 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 ...
0
votes
1answer
17 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 ...
1
vote
0answers
12 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 ...
-1
votes
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 $\|f\|_2 \le \|f\|_\infty$.
0
votes
2answers
21 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 ...
1
vote
0answers
10 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 ...
3
votes
2answers
34 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 ...
0
votes
1answer
21 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 ...
0
votes
1answer
22 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. ...
1
vote
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 $j:L\rightarrow B$ ...
1
vote
4answers
62 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?
0
votes
1answer
30 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.
1
vote
0answers
19 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 ...
0
votes
1answer
34 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)
0
votes
1answer
38 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 ...
0
votes
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 $B$ ...
0
votes
1answer
15 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?
0
votes
0answers
27 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 ...
-1
votes
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).
0
votes
0answers
24 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 ...
-1
votes
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)
0
votes
0answers
8 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) ...
1
vote
0answers
29 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, ...
1
vote
0answers
31 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
votes
0answers
24 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 ...
0
votes
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
1
vote
2answers
38 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 ...
0
votes
2answers
46 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
votes
1answer
47 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 ...
0
votes
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
votes
0answers
15 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 ...