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 ...

learn more… | top users | synonyms

0
votes
1answer
93 views

nilpotent - derivative - why is characteristic important?

Let $V$ be the space of all $f(t) \in K[t]$ with $\mathrm{deg} f \leq n-1$ and let $\psi: V \to V$ with $\psi(f) = f'$. Further $\mathrm{char}(K) = 0$. Then $\psi$ is nilpotent. Since one can take ...
0
votes
1answer
54 views

generalized eigenvectors - questions

We introduced generalized eigenvectors and then we were given this example: Given an endomorphism $\varphi: V \to V$ with $\chi_\varphi(t) = (-1)^n (t-\lambda)^n$, then $U(\lambda) = V$ since ...
1
vote
2answers
78 views

Checking diagonalizability of a given $2\times 2$ matrix

Let $A$ be the matrix $ A = \left( \begin{array}{cc} a & c\\ 0 & a \\ ...
2
votes
1answer
620 views

Fourier transform over a diagonal matrix

Let $F$ be a $100 \times 100$ DFT matrix, and $U$ be a diagonal matrix with its diagonal entries being all positive, denoted by $U=\mathrm{diag}(u_1, u_2,\cdots, u_{100})$. My question is: Under ...
1
vote
2answers
398 views

A problem with annihilators in infinite-dimensional vector spaces

It's a known fact that $\mathrm{Ann}(S^\circ)=S$, where $S$ is a subspace of a finite dimensional vector space $V$. I'll include the definitions for the sake of completeness, since $\mathrm{Ann}(S)$ ...
2
votes
1answer
302 views

Distance between points in a convex set and outside of a convex set.

Let $W$ be a set of points in $\mathbb{R}^n$. Let $C$ be the convex hull of the members of $W$. Is there a simple way of demonstrating that for any $x \in C$ and any $y \in \mathbb{R}^n \backslash C$, ...
3
votes
1answer
2k views

Matrix for rotation around a vector

I'm trying to figure out the general form for the matrix (let's say in $\mathbb R^3$ for simplicity) of a rotation of $\theta$ around an arbitrary vector $v$ passing through the origin (look towards ...
2
votes
1answer
157 views

Linear Mapping/Matrices Proof

At first look a rather logical question which has till date stumped many of us attempting to solve it. Hmm, hope you guys could offer some brain power here :) $A$ is a matrix from $\mathbb{R}^{2,2}$, ...
1
vote
2answers
104 views

Dimension of null space of a given problem

For any $n\in \mathbb{N}$, let $P_{n}$ denote the vector space of all polynomials with real coefficients and of degree at most $n$. Define linear transformation $T \colon P_n \rightarrow P_{n+1}$ by ...
3
votes
2answers
144 views

Finding the dimension of a given vector space

What is the dimension of the space of all $n \times n$ matrices with real entries which are such that the sum of the entries in the first row and the sum of the diagonal entries are both zero? ...
1
vote
2answers
223 views

Number of pairs linearly independent vectors in vector space of dimension 3

Suppose V is a real vector space of dimension 3. Then what will be the number of pairs of linearly independent vectors in V? Cani say it should be infinity? Because there exist infinite number of ...
2
votes
3answers
454 views

The ring $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$

The set $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ spans a ring under real addition and multiplication. Which elements have multiplicative inverses? This is part of an exercise from an introductory text ...
4
votes
2answers
2k views

Finding the dimension of real symmetric matrices with trace zero

What is the dimension of the vector space of all symmetric matrices of order $n\times n$ $(n\geq 2)$ with real entries and trace equal to zero?
4
votes
1answer
175 views

Is there a name for this $k$-fold vector product?

Let $V$ be a set of vectors of length $n$. Define a $k$-fold product on $V$, $$ \Upsilon(\{v_1,\ldots,v_k\}):=\sum_{j=1}^n\prod_{i=1}^k v_{ij}, $$ where $v_i\in V$ and $v_{ij}$ is the $j^\text{th}$ ...
8
votes
1answer
9k views

orthogonal eigenvectors

I have a very simple question that can be stated without proof. Are all eigenvectors, of any matrix, always orthogonal? I am trying to understand Principal components and it is cruucial for me to see ...
0
votes
1answer
66 views

orthogonal projection - simple exalanation needed

Could someone explain to me, using perhaps a very simple example in @d, what we mean by orthogonal projection from space D to space D'? Thanks
3
votes
2answers
115 views

How to find out the dimension of a given vector space?

What will be the dimension of a vector space $ V =\{ a_{ij}\in \mathbb{C_{n\times n}} : a_{ij}=-a_{ji} \}$ over field $\mathbb{R}$ and over field $\mathbb{C}$?
9
votes
1answer
2k views

How to count number of bases and subspaces of a given dimension in a vector space over a finite field?

Let $V_{n}(F)$ be a vector space over field $F=\mathbb Z_{p}$ with $\dim V_{n} = n$ i.e. cardinality of $V_{n}(\mathbb Z_{p}) = p^{n}$. What is the general criteria to find out the number of bases in ...
8
votes
1answer
205 views

Does there exist a vector space with 30 elements?

Does there exist a vector space with 30 elements? How to determine whether there exist any vector space of particular cardinality?
1
vote
1answer
75 views

Nonlinear system

We are given a non-linear system: $4x_1 − x_2 + x_3 = x_1x_4,$ $−x_1 + 3x_2 − 2x_3 = x_2x_4$ $x_1 − 2x_2 + 3x_3 = x_3x_4$ $x_1^2 + x_2^2 + x_3^2 = 1$ And the question asks: Show how to solve the ...
3
votes
1answer
306 views

Calculating the inertia of a real symmetric (or tridiagonal) matrix

I'm trying to find a quick method for evaluating the inertia of a real symmetric matrix, though I don't need to evaluate eigenvalues directly. The inertia of a matrix is a triple of the number of ...
2
votes
1answer
104 views

eigenvector computation

Given a full-rank matrix $X$, and assume that the eigen-decomposition of $X$ is known as $X=V \cdot D \cdot V^{-1}$, where $D$ is a diagonal matrix. Now let $C$ be a full-rank diagonal matrix, now I ...
1
vote
1answer
494 views

2x2 Matrix with real entries and a complex eigenvalue can't be normal

Homework question. Let $A$ be a $2 \times 2$ matrix with real entries. Suppose that $A$ has an eigenvalue $\lambda$ with the imaginary part of $\lambda \neq 0$. Is there an orthonormal basis of ...
3
votes
1answer
156 views

If $null(A) \subset null(B)$ can we draw any conclusion about range spaces of A and B

A and B are given $n\times$ m matrices If $null(A) \subset null(B)$ what conclusion can we draw about range Space of $A$ and $B$. Can we conclude that range space of B is contained in a range space of ...
4
votes
1answer
103 views

From $\dim A\leq \dim B$, can we conclude that $A\subseteq B $?

We have two subspaces $A$ and $B$ of a vector space $V$ such that $\dim A\leq \dim B$. Can we conclude that $A\subseteq B $ ? I need a proper justification.
0
votes
1answer
62 views

Tensor Product Question

For a finite dimensional vector space $V$, is it true that $\bigwedge^{n - 1}V \otimes V = \bigwedge^{n}V \oplus \ker(\bigwedge^{n - 1}V \otimes V \overset{\psi}{\rightarrow}\bigwedge^{n}V)$ where ...
1
vote
1answer
338 views

How to show that the range of a given matrix is contained in a given vector space?

$$B = \left(\begin{matrix} 0.4 & 0 & 0 & 0 \\ 0 & 0.4 & 0 & 0 \\ 0 & 0 & 0.4 & 0 \end{matrix}\right)$$ Here, $B \in \mathbb{C}^{3 \times 4}$ where $\mathbb{C}$ is ...
1
vote
1answer
2k views

determinant of a sum

I need a formula for the determinant of the sum of two matrices: $\det(\mathbb{I}+M)$. On the internet I found it for the first order but i need it at second or even third order. Where can I find the ...
5
votes
1answer
115 views

What is good about simple Lie algebras?

Recently I've been reading Naive Lie Theory by John Stillwell. In the book our aim usually concerns finding whether Lie algebras or Lie groups are simple. I wonder what beautiful properties does a ...
5
votes
0answers
165 views

Points and lines covering them

Let $n$ be a positive integer. A subset $S$ of points in plane satisfies the following conditions: a) We can't find $n$ lines in plane, such that every element of $S$ belongs to at least one of these ...
2
votes
3answers
220 views

Proposition about curves in $S^2$

Let $\gamma_1,\gamma_2:(a,b)\to S^2$ be unit speed curves in $S^2=\{\vec{v}\in\mathbb{R^3}:\vec{v}\cdot\vec{v}=1\}$. Then the following two statements are equivalent: (1) There is a $3\times 3$ ...
0
votes
1answer
159 views

One dimension subspaces of $V_{2}(q)$

Suppose that $V=V_{2}(q)$ is a vector space on a finite field $GF(q)$, so $|V|=q^{2}$. I saw this problem somewhere, " Describe one dimension subspaces of $V$ and find the number of them". What I ...
4
votes
3answers
1k views

Can the product of two non-zero symmetric matrices be anti-symmetric?

I'm trying to find an example to show that the product of two non-zero symmetric matrices can be anti-symmetric. I've proven that this is impossible for 2x2 matrices. For 3x3 matrices, I've ...
0
votes
0answers
58 views

geometrical interpretation of $\mathbb{Z}/2\mathbb{Z}$ graded space

According to wikipedia, a $\mathbb{Z}/2\mathbb{Z}$ graded space (super vector space) $V$ is a a vector space which can be decomposed in a direct sum $V=V_0 \oplus V_1$ where elements of $V_0$ are ...
1
vote
3answers
288 views

Show: If the adjoint of T is -T, all eigenvalues are purely imaginary

Homework question. Let $V$ be a finite dimensional inner-product space over $\mathbb{C}$. Let $T \in L(V,V)$ satisfy $T^*=-T$. Show that all eigenvalues of $T$ are purely imaginary, i.e., if ...
0
votes
1answer
102 views

Eigenvectors and Eigenvalues [duplicate]

Possible Duplicate: “Eigenrotations” of a matrix have a question: If a matrix $M$ acts by stretching a vector $x$ not changing its direction, then $x$ is an eigenvector of $M$. ...
2
votes
0answers
69 views

Eigenvalue of a form

I came across the following matrix while reading an article..Can you please help me to understand the following. We are defining following form: ...
3
votes
1answer
963 views

Formula for cylinder

In an exercise I was asked to find a formula of the form $F(x,y,z)=C$ for a cylinder though the axis $(t,t,t)$ and radius $R$. The formula I got seemed a bit suspicious so I wanted to ask if I have it ...
2
votes
1answer
302 views

Subspace of V is also a null space of T

Prove that any subspace of vector space $V$ is a null space over some linear transformation $V \rightarrow V$. So far I have: Let $W$ be the subspace of $V$, let $(e_1, e_2, \ldots, e_r)$ be the ...
2
votes
2answers
512 views

How to find the orthogonal projection of a vector over a unit ball?

I have unit balls defined by the $1$, $2$ and $\infty$ norm in $\mathbb{R}^2$. I want to find the orthogonal projection of a vector $(x,y)$ onto the balls. How could it be done? I only know how to ...
0
votes
1answer
35 views

what should be the frequency distribution of the eigenvalues of a randomly generated hermitian matrix?

I'm getting the eigenvalues of a randomly generated hermitian matrix distributed like a normal probabilistic distribution(crowded in the middle values ) but my sir told me that it should be a ...
0
votes
1answer
40 views

Path contained in Surfaces

$y(t)$ is a path contained in two surfaces: $x^2+y^4+z^6=3$, $x+y^2=y+z^2$ also $y(0)=(1,1,1)$ and $||y'(0)||=1$ Need to find the vectors $-y'(0)$ and $+y'(0)$ To be honest, I'm not sure how to ...
1
vote
2answers
142 views

compositions of permutations

For compositions of permutations on a set $X = \{1,2,3\}$, my lecture notes say that the composition $\phi_2 \phi_1$ is the permutation $\phi_1$ followed by the permutation $\phi_2$. So consider the ...
13
votes
1answer
402 views

Eigenvalues for $3\times 3$ stochastic matrices

This is a plot of the non-real eigenvalues of 10000 randomly generated $3\times3$ stochastic matrices. It's pretty clear that they lie in the convex hull of the three cube roots of unity. The ...
1
vote
0answers
134 views

Calculation of stopping condition for Conjugate Gradient

I am a person with programming background and need some math help. I am looking at the source code for an implementation of the Conjugate Gradient iterative solver ...
1
vote
1answer
91 views

Multiplying polynomials

Let $f(x)$ be degree $n$ polynomial, with $n+1$ nonzero monomial, i.e., all coefficients nonzero (for example if $n = 3$, then we could have $3x^3 + 2x^2 + x + 10$) Let $g(x)$ be any polynomial of ...
0
votes
1answer
77 views

Can I make my checks for Bilinear forms shorter?

First I'll define what I talk about: A bilinear form on a vector space V is a mapping: $F: V \times V \rightarrow \mathbb{R}, (a,b) \mapsto F(a,b)$ which is linear in every argument: $a, b, c \in ...
0
votes
1answer
193 views

Determine linear operator image?

I have a simple linear operator: $$\begin{align}g: \Bbb{R^4} &\to \Bbb{R^3}\\g (x, y, u , v) &= ( x + u, x + v, y + u)\end{align}$$ How would I determine the image of this linear ...
1
vote
1answer
349 views

The number of subspaces of a vector space

Let $V$ be a vector space of dimension $n$ over $\mathbb{F}_q$, and let $U$ be a subspace of dimension $k$. I want to compute the number of subspaces $W$ of $V$ of dimension $m$ such that $W\cap U=0$. ...
3
votes
1answer
107 views

Polynomial with a root that occurs n times, the root must be 0

I'm having a bit trouble with this excercise: The problem: Let there be a polynomial $f(x)=a_1x^{t_1} + a_2x^{t_2} + ... + a_nx^{t_n}$ Where $t_1, t_2, ..., t_n$ are not-negative integers. The ...