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

6
votes
2answers
109 views

let $A,B\in M_{n}(C)$ such that c is complex field and $AB^2-B^2A=B$ how prove $B^n=0$

Let $A,B\in M_{n}(C)$ such that $C$ is complex field and $AB^2-B^2A=B$. How prove $B^n=0$. thanks in advance
0
votes
3answers
80 views

Matrix with linearly dependent columns

Is there an example of a $3\times 3$ matrix $M = [v_{1}\, v_{2}\, v_{3}]$ (where $v_{i}$ are $3 \times 1$ vectors) such that $v_{1}, v_{2}, v_{3}$ are linearly dependent however if we have $av_{i} + ...
3
votes
1answer
81 views

$P_{1c} = AP$ , $P_{2c} = BP$. How to find $P$? (being that $A$ and $B$ are $3\times 4$ matrices and $P$ is a $4\times 1$ vector)

This problem arose in my stereo vision project. $$ P_{1c} = A*P $$ $$ P_{2c} = B*P $$ where: $P_{1c}$ and $P_{2c}$ are $3\times1$ vectors, $A$ and $B$ are $3 \times 4$ matrices and $P$ is a ...
1
vote
1answer
277 views

What is the dimension of the intersection of hypersurfaces in an infinite dimensional vector space?

Let V be a vector space of dimension $\text{dim}\left(V\right)=k$, where $k$ can be $\infty$. If I have two hypersurfaces in V of dimensions $\text{dim}\left(S_1\right)=l$ and ...
1
vote
2answers
94 views

We can't give the structure of linear space on $\mathbb{Q}(\sqrt{2})$ for a set of rational numbers

Please help me prove, that we can't give the structure of a linear space over $\mathbb{Q}(\sqrt{2})$ for a set of rational numbers with standard addition operation. $\mathbb{Q}(\sqrt{2})$ is a field ...
3
votes
3answers
99 views

Sum of subspaces is a subspace

I've found today a task: Let $A$ and $B$ are subspaces of $V$. Prove, that $A\cup B$ is a subspace only, if $A\subset B$ or $B \subset A$. Please explain, why my counterexample isn't correct. ...
4
votes
3answers
157 views

Find $C$, if $A=CBC$, where $A$,$B$,$C$ are symmetric matrices.

If $A=CBC$, where $A$,$B$,$C$ are symmetric matrices and $A$,$B$ are given find $C$. $A$,$B$,$C$ are assumed to be real valued and $B$ is positive definite matrix. Does the unique solution always ...
0
votes
1answer
113 views

Looking for not/continuous and differentiable function examples?

I have to give an example of functions which are: continuous and not differentiable $f(x)=|x|$ differentiable $f(x)=(1/2)* x *|x|$ not continuous $f(x)=1/x$ and $f(x)=1/cos(x) [0;\pi]$ Are ...
3
votes
2answers
308 views

A standard quadratic minimization problem

Consider the "Complex" Quadratic minimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~\mathbf{{x}}^H\mathbf{Q}\mathbf{x}-2~\Re{(\mathbf{x}^H\mathbf{b})}+1 \end{align} ...
2
votes
2answers
65 views

Method for finding the number of matrices in $M_2(\mathbb{Z}_3)$ whose determinant is $1$

I just came across a question which asks to find all the $2 \times 2$ matrices in $\mathbb{Z}_3$ whose determinant is $1$. I know that since there are only three elements in $\mathbb{Z}_3$, it is ...
2
votes
1answer
76 views

How can I break down a rotation of known amount around a known axis into two rotations of unknown amounts around known axes?

I have a vector that's been rotated a known amount about a known axis. I would like to break this rotation down into two separate rotations around known, linearly independent axes where the amounts I ...
1
vote
1answer
136 views

LU decomposition permutation matrix

Hi can you help me with the following; Let $A$ be an $n\times n$ matrix and have $LU$ decomposition with lower and upper triangular matrices. Let $P =\{e_n,e_{n-1},\ldots,e_1\}$ where $e_i$ is a ...
5
votes
1answer
58 views

Approximate a function using another function

Problem Find the best approximation of $f(t)=t^2$ with $h(t)=ae^t+be^{2t}+c$ everywhere on the interval $[0,4]$. Attempt I know how to solve this problem given sample points, by using least squares, ...
0
votes
0answers
118 views

Solving systems of linear congruences with rational coefficients

Is there any way to solve for $x$ in a system of linear congruences with rational coefficients in the following form? $$Ax \equiv b\pmod 2, \space where\space A \in \Bbb Q^{n,m}, b \in \Bbb Q^m$$ ...
0
votes
0answers
48 views

Convergence of a sequences of projecting point on a set of hyperplanes

How to prove next statement? Consider finite number intersecting hyperplanes in N dimensional Euclidean space and some starting point $x_0$. Point $x_i$ is obtained from point $x_{i-1}$ by projecting ...
0
votes
0answers
42 views

Matrix equivalence relations [duplicate]

Possible Duplicate: Equivalence Relation? Column-equivalence on two $m\times n$ matrices. If $A\sim B$, how do I show that $\sim$ is an equivalence relation for all $m\times n$ matrices. ...
0
votes
0answers
120 views

Showing two operations make a vector space

I'm asked to show that { $f:\mathbb R \rightarrow \mathbb R | f$ continuous} with operations (a) $(f+g)(t)=f(t)+g(t)$ (b) $(rf)(t) = rf(t)$ make a vector space. I'm trying to show (i)-(iii) below, ...
1
vote
3answers
970 views

Solving system of linear differential equations by eigenvalues

Using eigenvalues and eigenvectors solve system of differential equations: $$x_1'=x_1+2x_2$$ $$x_2' = 2x_1+x_2$$ And find solution for the initial conditions: $x_1(0) = 1; x_2(0) = -1$ I tried to ...
0
votes
1answer
23 views

Interesting question regarding linear dependency

if I have five linearly dependent vectors in the vector space $\mathbb{R}^5$, lets call them $A_i$ for $i=1$ to $5$. can I find $C_{i,j}$ for $i=1,...,5$ and $j=1,...,5$ so that I get five new vectors ...
4
votes
2answers
75 views

how prove for n≥2 $0≤\sum_{i=1}^n \sum_{j=1}^na_{ij}≤n $ that $A=[a_{ij}]\in M_n(R)$ be real matrix that A is symmetric and idempotent

let $A=[a_{ij}]\in M_n(R)$ be a real, symmetric and idempotent matrix (i.e.$A^2=A$). how can we prove for $n \geq 2$ that $$0≤\sum_{i=1}^n \sum_{j=1}^na_{ij}≤n $$ thanks in advance.
0
votes
1answer
78 views

Bounded solution for positive-definite matrix

Suppose $A$ is a positive-definite matrix and $b$ is a vector which satisfies $ b\leq \mbox{diag}(A)$ for all entries of $b$, i.e. $b_i= b^T e_i\leq e_i^T A e_i $. The linear equations holds: $Ax=b$ ...
0
votes
1answer
216 views

Find lower triangular matrix using Schur Factorization

I need to find lower triangular matrix using Schur Factorization $A' = U^T A U$ . Actually after factorizing it results upper triangular matrix [using MATLAB] Expecting result could be as such $$A'= ...
3
votes
0answers
91 views

What breaks down in linear algebra over the rings (or commutative rings)? [duplicate]

Let $R$ be a ring (with $1$). Last night, I was trying to prove that $M_{n}(R)$ (the ring of $n \times n$ matrices over $R$) is a ring. As I have done in my previous linear algebra course (which was ...
1
vote
0answers
85 views

Gamma Matrices in Four dimensions

Let $\gamma^0$, $\gamma^1$, $\gamma^2$ and $\gamma^3$ be the Gamma matrices in four dimensions. My question is, how do I convince myself that these four matrices generate the set of 4-by-4 matrices ...
6
votes
1answer
221 views

Question from Artin's algebra book

Let $A$ be an $n\times n$ matrix such that $A^r =I$ and $A$ has exactly one eigenvalue ,then $A= \lambda I$. My answe: As $A$ is a $n\times n$ matrix then characteristic polynomial has degree n and ...
0
votes
1answer
309 views

Orthogonal matrix Q of A such that $Q^T A Q$ is a diagonal matrix

Given a matrix $A$, how could I find an orthogonal matrix $Q$ such that $Q^t A Q$ is a diagonal matrix?
1
vote
1answer
56 views

Example where $P(n)$ is not valid for induction?

I am studing for my math exam on friday and at the moment I am doing a examples about induction: However, I am struggeling at this question: Give an example where $P(n)$ is not valid even if the ...
0
votes
2answers
207 views

Calculating Diagonal Matrix, too many zeroes in the eigen vectors, what now?

Given the Matrix $$A = \left(\begin{matrix} 1 & 1 & 0 \\ 0 & 2 & 0 \\ 1 & 0 & 1 \end{matrix}\right)$$ calculate the diagol matrix $diag(A)$ Well, for this I need the ...
1
vote
1answer
95 views

Is a symmetric non-negative integral matrix with odd diagonal entries and even non-diagonal entries full rank over $\mathbb{R}$?

Let $A$ be an $n\times{}n$ matrix which satisfies the following properties: The elements of $A$ are non-negative integers. The diagonal elements of $A$ are all odd. The non-diagonal elements of $A$ ...
0
votes
0answers
452 views

Determinant of a symmetric matrix values in each column and row don't repeat

Could you help me count the determinant of this symmetric matrix? $\begin{vmatrix} a&2b&3c&6d\\b&a&3d&3c\\c&2d&a&2b\\d&c&b&a\end{vmatrix} $
1
vote
2answers
148 views

Partial Derivatives involving Vectors and Matrices

Let $Y$ be a $(N \times 1)$ vector, $X$ be a $N \times M$ matrix and $\lambda$ be a $M \times 1$ vector. I am wondering how I can evaluate the following partial derivative. \begin{align} ...
3
votes
1answer
228 views

fixed point projective geometry

I am thinking about the following: Let $\sigma:\mathbb C P^n\rightarrow\mathbb C P^n$ be a projectivity with $\sigma\circ\sigma=id_{\mathbb C P^n}$. I define the set of all fix points by ...
0
votes
1answer
125 views

Homogeneous system of linear equations over $\mathbb{C}$

I have two systems of linear equations and I need to verify if they are indeed the same system, and if they are I must rewrite each equation as a linear combination of the others. A: ...
0
votes
0answers
138 views

Linear Algebra: equivalence classes

Give the corresponding equivalence classes for $3\times2$ matrices over the field $\mathbb{Z}_5$ for row and column equivalence classes. Whenever $B$ can be obtained from $A$ by performing a ...
1
vote
1answer
89 views

Calculate $\dim W+V$ and $W\cap V$

This is a task from an old exam. Let define: $V_{t} = \text{lin}((1,2,2,1),(1,1,-1,t))$ $W = \cases{x_1-x_2=0\\x_3-x_4 =0}$ Calculate $\dim W+V_{t}$ and $\dim W\cap V_{t}$ Please verify answer ...
1
vote
1answer
63 views

How to calculate symmetry

This is a task from an old exam. Let define: $V = \text{lin}((1,2,2,1),(1,1,-1,1))$ $W = \cases{x_1-x_2=0\\x_3-x_4 =0}$ $\varphi:\mathbb{R}^{4}\to \mathbb{R}^{4}$ - a symmetrion against V along ...
1
vote
1answer
143 views

fitting two parallel lines to two clusters of points

I have a problem where I have two clusters of 3D points and I am trying to find the two parallel lines that are a given distance $d$ (i.e $d$ is not a variable) apart that will be a best fit for the ...
2
votes
1answer
248 views

System of Linear Equations using Mod

I just want to check that I did a certain problem correctly. This is it: $$a+b=3 \pmod{26}\\2a+b=7 \pmod{26}$$ Solve for $a$ and $b$ Now I setup the augmented matrix: $$\left[ \begin{array}{ccc} 1 ...
1
vote
2answers
489 views

How to prove that the Gram matrix $A^TA$ is symmetric with nonnegative diagonal elements?

If A is any real $m \times n$ matrix, then $A^TA$ is called the Gram matrix of $A$. Show that the Gram matrix of $A$ is always symmetric with nonnegative diagonal elements. I have tried several ...
0
votes
1answer
260 views

How to Calculate the Direction of a Vector

Let's say I have points $a$, $b$, and $c$. We also have $\vec{ab}$ and $\vec{ac}$. Finally, we know neither vector's direction. (That is, the vector's angle on each axis as if the vector were ...
5
votes
1answer
154 views

Use row reduction to prove that $\det(\mathbf{A})=\det(\mathbf{A}^{T})$

I need to prove that the determinant of a matrix is equal to the determinant of its transpose. This fact is obviously easy to prove using the definition of the determinant, but the question stipulates ...
1
vote
1answer
97 views

Find a $3\times 3$ matrix $X$, such that $X^{3}$ = specific matrix [duplicate]

Possible Duplicate: Given a matrix $A$ find a matrix $C$ such that $C^3$=$A$ I have stumbled upon the following question while studying for a test in linear algebra: Find a matrix $X $ of ...
1
vote
0answers
155 views

Centre of a spherical triangle

Suppose I have a triangle defined by 3 unit vectors {$v_1, v_2, v_3$} in a 3 dimensional complex inner product space. What would be the centre of such a triangle? I guess it should be something like ...
3
votes
2answers
165 views

$A$ be a $n \times n$ matrix with $\text{rank}\,(A)\lt n$: multiple choice question

Let $A$ be an $n \times n$ real non-zero matrix of rank less than $n$. Then one of the following is true? : (A) there exists an $n \times n$ real non-zero matrix $B$ such that $BA = 0$. ...
0
votes
0answers
83 views

Can antiunitary symmetry be used to calculate determinant of a matrix

Suppose I have some $N \times N$ complex matrix $A$, that commutes with some antiunitary operator $U$ that satisfies $U^2 =-1$. It can be shown that $\det(A)\ge 0$ , because for every eigenvector ...
4
votes
1answer
101 views

Saturating Horn's Inequalities

If I have a matrix product of the form: $C = AB$ where $A = UDU^*$ With A and B square, Hermitian and positive semidefinite, D diagonal, U a unitary and * representing the conjugate transpose, ...
5
votes
3answers
122 views

Problem related to a square matrix

Let $A$ be an $n\times n$ matrix with real entries such that $A^{2}+I=\mathbf{0}$. Then: (A) $n$ is an odd integer. (B) $n$ is an even integer. (C) $n$ has to be $2$ (D) $n$ could be any positive ...
4
votes
2answers
117 views

How should I prove a set is convex?

Given a set $$ \mathbf{S} = \{ \mathbf{x}\:|\: \mathbf{x}^T\mathbf{V}\mathbf{x}=1 \} $$ where $\mathbf{V}$ is a positive semidefinite matrix. How to prove this set is convex? I tried in the ...
3
votes
3answers
120 views

The expression of $f(x,y)$ explicitly

Consider the function $f : \mathbb{R^2} \rightarrow \mathbb{R^2}$ that associates to each vector $(x,y)$ the vector $f(x,y)$ obtained from a counterclockwise rotation of $\theta=30^{\circ}$ of the ...
6
votes
3answers
573 views

Eigenvector and its corresponding eigenvalue

For the following square matrix: $$ \left( \begin{array}{ccc} 3 & 0 & 1 \\ -4 & 1 & 2 \\ -6 & 0 & -2 \end{array} \right)$$ Decide which, if any, of the ...