Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

Linear algebra is concerned with vector spaces and linear transformations between them:

$$(x_1, \dots, x_n)\to a_1x_n+\dots+a_nx_n$$

Concepts include systems of linear equations, bases, dimensions, subspaces, matrices, determinants, kernels, null spaces, column spaces, traces, eigenvalues and eigenvectors, diagonalization and Jordan normal forms.

This is a general tag, most of the subjects included have secondary tags, e.g.

127034 questions
5
votes
1 answer

Linear map $f:V\rightarrow V$ injective $\Longleftrightarrow$ surjective

Maybe I am not good at looking for the right questions but I haven't seen this task anywhere so I hope it is no duplicate. I have to prove the following statement: Let $V$ be a finite dimensional vector space and $f:V \rightarrow V$ a linear map.…
user114193
5
votes
1 answer

Is a diagonal matrix diagonalizable?

A matrix $A$ is a diagonalizable if there exists a diagonal matrix $D$ such that $A$ is similar to $D$. If $A$ is a diagonal matrix, though, is it diagonalizable? If so, it would seem $D$ would just be $A$. I suppose my real question is if it is…
Cormano
  • 51
5
votes
1 answer

matrices - traces and square root

I'm trying to show that Tr$\left(\sqrt{(\mathbf{PXP}^\top)}\right) \le \text{c Tr}\left(\mathbf{P}\sqrt{\mathbf{X}}\mathbf{P}^\top \right)$ where Tr is the trace operator, $\mathbf{X}$ is symmetric positive semi-definite matrix, $\mathbf{P}$ is a…
user113440
  • 51
5
votes
2 answers

Find all 4x4 A matrices so that $A^4=A^6$

Find all 4x4 A matrices so that $A^4=A^6$. I think the method has to do something with eigenvalues, eigenvectors etc'... Thanks in advance for any assistance!
err
  • 141
5
votes
1 answer

Similar matrices and eigenvalues

Given two invertible matrices $A,B\in M_{2}(\mathbb R)$ such that $B^{-1}AB=A^{2}$, and $1$ is NOT an eigenvalue of $A$. (1) Find the eigenvalues of A, and (2) Find $A$ and $B$ satisfying the given conditions. What I tried is as follows: since …
William
  • 61
5
votes
2 answers

Simultaneous diagonalizability of commuting unitary operators

I'm trying to prove the following: If $S\colon V\to V$ and $T\colon V\to V$ are unitary linear transformations on unitary space $V$ ($\dim V=n$, $n$ is finite), such that $ST=TS$, then they have a joint eigenvector basis (aka there is a basis of…
iroiroaru
  • 51
5
votes
2 answers

Prove that if $g(t)$ is relatively prime to the characteristic polynomial of $A$, then $g(A)$ is invertible

I'd like to write to you two problems that I tried to solve, I'm not sure of the solution of the first. Let $A\in M_n (F)$ be a matrix and $g(t)\in F(t)$ a polynomial, $P_A(t)$- the characteristic poly of $A$. I need to prove that if…
user6163
5
votes
2 answers

Why are these examples striking?

The question is from an exercise in Gilbert Strang's Linear Algebra and its Applications. The powers $A^k$ approach zero if all $|\lambda_i|<1$, and they blow up if any $|\lambda_i|>1$. Peter Lax gives four striking examples in his book Linear…
user9464
5
votes
2 answers

Change of basis matrix for orthogonal bases

I am trying to show that if $B_1$ and $B_2$ are orthonormal bases for $\mathbb{R}^n$, then the change of basis matrix $P$ from $B_1$ to $B_2$ is an orthogonal matrix. I'm a bit stuck. I started with this: Let $x,y \in \mathbb{R}^n$. Then…
Alex137
  • 51
  • 1
  • 3
5
votes
4 answers

Prove that $\sin(x + \alpha)$, $\sin(x + \beta)$ and $\sin(x + \gamma)$ are linearly dependent.

Functions $f$ and $g$ are independent on an interval $D$ if $af(x) + bg(x) = 0$ implies that $a = 0$ and $b = 0$ $\forall x \in D$ let $\alpha$, $\beta$, $\gamma$ be real constants. Prove that $\sin(x + \alpha)$, $\sin(x + \beta)$ and $\sin(x +…
Zhoe
  • 2,415
5
votes
1 answer

Polynomials and Linear Operators

Let $p$, $q$, and $r$ be polynomials such that $p(x) = q(x)r(x)$, and let $T$ be a linear operator on a vector space $V$. Is there a simple way to show that $p(T) = q(T)r(T)$ ?
guest
  • 51
5
votes
3 answers

A vector that is orthogonal to the null space must be in the row space

Simple question. We know from the fundamental theorem of linear algebra that the nullspace of a matrix is the orthogonal complement of its row space. I can write this as: Let $M$ be a matrix. The following two conditions are equivalent: (i) $u$ is…
user46234
5
votes
1 answer

An exercise for eigenvalues and eigenvectors

The following is from an exercise in Gilbert Strang's Linear Algebra and its Applications: Suppose $A$ has eigenvalues $0,3,5$ with independent eigenvectors $u,v,w$. Find a particular solution to $Ax = v+w$. Find all solutions. It is not…
user9464
5
votes
5 answers

Definition of Vector Space

What is the meaning of objects in a vector space? Definition: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars, subject to the ten axioms... Can…
Dante
  • 1,908
5
votes
1 answer

Orthogonal projection and two subspaces

Let $\mathcal{S}$ and $\mathcal{T}$ be two subspaces of $\mathbb{R}^n$, let $P$ be the orthogonal projection of $\mathbb{R}^n$ on $\mathcal{S}$ and let $Q$ be the orthogonal projection of $\mathbb{R}^n$ onto $\mathcal{T}$. Show that if $P$ and $Q$…
Mark
  • 365
  • 5
  • 13
1 2 3
…
99 100