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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Can we say that there exist an integer n such $A+nB$ invertible?

If $A$ and $B$ are $3\times 3$ matrices and $A$ is invertible, then can we say that there exist an integer $n$ such that $A+nB$ invertible? I was trying by choosing n such that eigne values of $A+nB$ ...
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1answer
402 views

Symmetric power of vector space

Let $V$ be a vector space over a field $k$ of char. zero and denote by $Sym^n_k V$ its $n$-th symmetric power over $k$. Now I simply want to know what $Hom_k(V,Sym^n_k V)$ is for $n \geq 2$. To be ...
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votes
3answers
245 views

How does composition affect eigendecomposition?

What relationship is there between the eigenvalues and vectors of linear operator $T$ and the composition $A T$ or $T A$? I'm also interested in analogous results for SVD.
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votes
2answers
185 views

$2\times 2 $ matrices over $\mathbb{C}$ that satisfy $A^3=A$

Let $A$ be a $2\times 2$ matrix with complex entries. What would be the number of $2\times 2$ matrices $A$ that satisfies $A^{3} = A$. Question was are they infinite? If it is $3\times 3$ matrix then ...
3
votes
2answers
150 views

simplify this expression

I want to know how to simplify the following expression by using the fact that $\sum_{i=0}^\infty \frac{X^i}{i!}=e^X$. The expression to be simplified is as follows: $$\sum_{i=0}^{\infty} ...
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vote
3answers
169 views

Proof: $\ker \psi \subseteq \ker (\psi \circ \psi)$

Hi guys I'm completely clueless about this proof I came across in a uni textbook. Let $V$ be a vector space, and $\psi:V\to V$ be a linear transformation. Prove that $\ker \psi \subseteq \ker (\psi ...
4
votes
1answer
199 views

$A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to 1. find the possible value of the determinant

Let $A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to $1$. And remaining nine entries of the row being $0$. Which of the following is not a possible value of the ...
3
votes
1answer
68 views

solution for an equation

I have two diagonal matrices over the reals, both contain positive values on the diagonal (and 0 everywhere else), $A$ and $B$. $A$ is of size $n \times n$ and $B$ is of size $m \times m$. We assume ...
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3answers
193 views

Some properties of a $2\times 2$ matrix with repeated eigenvalues

I got a problem in my exam Consider the matrix $ A =\left( \begin{array}{cc} a & b \\ c & ...
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1answer
282 views

What is this linear operator/matrix?

I have a linear operator with its matrix in certain coordinates to be $$ \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & \frac{1}{2} & 0 ...
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votes
2answers
368 views

Nilpotent matrix question

If $N$ is an $n\times n$ nilpotent matrix such that $N^k=0$ for some integer $k$. Is it true that $(DN)^k=0$ for any diagonal matrix $D$?
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115 views

Counting number of linear transformations

Let $v_{1} = (1, 0)$, $v_{2} = (1, -1)$ and $v_{3} = (0, 1)$. How many linear transformations $T :\mathbb {R^2}\rightarrow \mathbb {R^2} $ are there such that $T(v_{1} ) = v_{2}$, $T(v_{2} ) = ...
0
votes
1answer
103 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
57 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 ...
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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
795 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 ...
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2answers
525 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
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1answer
370 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 $y \in \mathbb{R}^n \backslash C$, ...
4
votes
1answer
3k 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
162 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 $\mathbb{R}^{2,2}$, ...
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2answers
109 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
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2answers
151 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? ...
2
votes
2answers
274 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
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3answers
586 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 ...
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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
200 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 $j^\text{th}$ ...
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1answer
12k 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
70 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
120 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 $\mathbb{C}$?
12
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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
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1answer
212 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?
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1answer
83 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
439 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
109 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 ...
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1answer
603 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
165 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
105 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
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1answer
64 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
434 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 ...
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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
122 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
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0answers
166 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 ...
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votes
3answers
222 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 $3\times 3$ ...
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1answer
189 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
2k 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 ...
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0answers
63 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 ...
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3answers
361 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 ...
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1answer
111 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 $M$. ...
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71 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: ...
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1answer
1k 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 ...