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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
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Prove if $A \in Mat_{n,n}(\mathbb F)$ is both symmetric and skew-symmetric then $A=0$

Prove if $A \in Mat_{n,n}(\mathbb F)$ is both symmetric and skew-symmetric then $A=0$ I know $A^T = A = -A \Rightarrow A = -A \Rightarrow A_{i,j} = -A_{i,j}$. Since $\mathbb F$ is a field we have $2A_{i,j} = 0 \Rightarrow 2 = 0 \lor A_{i,j} =…
Shuzheng
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Describing a linear map geometrically

I have the following linear map $\mathbb{R}^2\to\mathbb{R}^2:$ $$\begin{pmatrix}x\\y\end{pmatrix}\mapsto \begin{pmatrix}9y-5x\\7y-4x\end{pmatrix}$$ I am asked to describe this geometrically. I can see what is happening: the upper left quadrant is…
user118224
  • 1,509
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Solve for $X$ in $Y = X^TAX$.

Suppose $$ Y = X^TAX, $$ where $Y$ and $A$ are both known $n\times n$, real, symmetric matrices. The unknown matrix $X$ is restricted to $n\times n$. I think there should be at least one real valued solution for $X$. How do I solve for $X$? …
Andrew H
  • 205
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Prove $\| A(A^TA)^{-1}A^T\|_2 = 1$ when rank of matrix $A$ is $n$

Given a matrix $A \in R^{m \times n}$ and whose rank is $n$. I need to show $\| A(A^TA)^{-1}A^T\|_2 = 1$. Can any hint me the direction in which I should solve this problem. Should I use any decomposition of matrix $A$ to show the result?
Learner
  • 2,696
8
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There exists $C\neq0$ with $CA=BC$ iff $A$ and $B$ have a common eigenvalue

Question: Suppose $V$ and $W$ are finite dimensional vector spaces over $\mathbb{C}$. $A$ is a linear transformation on $V$, $B$ is a linear transformation on $W$. Then there exists a non-zero linear map $C:V\to W$ s.t. $CA=BC$ iff $A$ and $B$ have…
NGY
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Why invariance to change of basis is so important in linear algebra?

I'm reading a book on linear algebra and I see that for every new presented concept (from simple vectors and linear functions and up to tensors) we immediately study how does it behave under a change of basis. Is it invariant or not, etc. This idea…
lithuak
  • 1,205
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$A\in M_n(\mathbb C)$ invertible and non-diagonalizable matrix. Prove $A^{2005}$ is not diagonalizable

$A\in M_n(\mathbb C)$ invertible and non-diagonalizable matrix. I need to prove that $A^{2005}$ is not diagonalizable as well. I am asked as well if Is it true also for $A\in M_n(\mathbb R)$. (clearly a question from 2005). This is what I did: If…
Jozef
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3 answers

Existence theorem about the adjoint.

I'm trying to prove the existence of the linear map $f^*$ in the following theorem about the adjoint: Let $f: V \to W$ be a linear map, with $V$ finite-dimensional and $V,W$ inner product spaces over the same field. Then there is a unique linear map…
Mussé Redi
  • 1,429
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3 answers

$A^2=I,\det A>0$ implies $A+I$ is non-singular

Question: If a square matrix $A$ satisfies $A^2=I$ and $\det A>0$, show that $A+I$ is non-singular. I have tried to suppose a non-zero vector $x$ s.t. $Ax=x$ but fail to make a contradiction. And I tried to find the inverse matrix of $A+I$ directly,…
NGY
  • 1,027
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Commutation when minimal and characteristic polynomial agree

Hello I am studying for the qualifying exam in algebra and I am having trouble solving this seemingly easy problem. If $A$ is a matrix whose minimal polynomial and characteristic polynomial agree, and $B$ commutes with $A$ then $B$ is a polynomial…
Joe
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Minimal polynomial and diagonalizable matrix

Let $B,C$ square matrices above a field,and $D$ Rectangle matrix in the correct size above the same field. $A=\begin{pmatrix} B &D \\ 0& C \end{pmatrix}$ I need to prove that if $A$ is Diagonalizable so Do $C$ and $B$, And if $B$ and $c$ are…
user6163
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1 answer

How to prove that this $X_0$ is nilpotent

Let $V=M_n(\mathbb C)$ and $A\subseteq B$ are subspaces of $V$. Let also $$ M=\{X\in V:\ \forall Y\in B,\ XY-YX\in A\}. $$ Suppose $X_0\in M$ enjoys the property that $\operatorname{tr}(ZX_0)=0$ for any $Z\in M$. Show that $X_0$ is nilpotent. My…
user94270
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4 answers

A $2 \times 2$ matrix $A$ such that $A^n$ is the identity matrix

So basically determine a $2 \times 2$ matrix $A$ such that $A^n$ is an identity matrix, but none of $A^1, A^2,..., A^{n-1}$ are the identity matrix. (Hint: Think geometric mappings) I don't understand this question at all, can someone help please?
user1952811
  • 205
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Prove that $\{ \sin(x), \sin^2(x), \sin^3(x)\}$ is linearly independent in $F(\Bbb R)$

Prove that $\{ \sin(x), \sin^2(x), \sin^3(x)\}$ is linearly independent in $F(\Bbb R)$. I tried plugging in $\left\{ 0, \frac {\pi} {2}, \pi, \frac {3\pi}{2}\right\}$ but that doesn't work. How should I prove this?
user95087
  • 629
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Geometric intuition of adjoint

For a linear operator it holds that $\ker (T^\ast ) = (\operatorname{ran} (T))^\perp$. The star denote the adjoint of $T$ and $\perp$ the orthogonal complement. Is there a geometric intuition for the meaning of $\ker (T^\ast ) = (\operatorname{ran}…
blue
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