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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 null $T^k$ = null $T$ and range $T^k$ = range $T$

I'm trying to prove that if $T$ is a normal operator, then null $T^k$ = null $T$ and range $T^k$ = range $T$. Showing null $T$ $\subset$ null $T^k$ is simple, so I'm working on the other inclusion. So far I've been able to deduce that for a vector…
Danny
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Automorphism on a finite dimensional vector space

If $V$ is a vector space over $\mathbb Q$ with $\operatorname{dim}(V)=3$. How we can prove that there is NO automorphism $\phi: V \rightarrow V$ such that $\phi^{-1}=2\phi$. I tried: Let $\phi: V \rightarrow V$ is an automorphism such that…
Emma
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Show that $\dim(U + V) = \dim(U) + \dim(V) - \dim(U \cap V)$

Let $W$ be a vector space and let $U$ and $V$ be finite dimensional subspaces. Not sure how to go about solving this.
user34166
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Is A diagonalizable?

$$A=\left(\begin{array}{ccccc} 1 &1 &\cdots &1 &1 \\ 1 &0 &\cdots & 0 & 1\\ \vdots & & \ddots & & \vdots\\ 1 & 0 & \cdots & 0 &1 \\ 1 &1 &\cdots &1 &1 \end{array}\right)\in M_{n}(\mathbb{R})$$ It has 1's around it and 0's…
user6163
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What's a non-zero (column) vector?

This is a basic question, but I can't find the definition on wikipedia, google, or math.stackexchange, because I only find examples of it being used in problems. Therefore, I want to clarify: Does a non-zero vector have at least one non-zero entry…
Rich
  • 367
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4 answers

characterize all matrices $X$ such that $BA = X$ whenever $AB = X$

It is clear that if $A$ and $B$ are $n\times n$ matrices (over a field) with $AB = I$ then $BA = I$. I like to characterize all matrices $X$ such that $BA = X$ whenever $AB = X$.
kian
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Congruent and Similar Matrices

I tried to solve the following question: Find 2 matrices A and B in M2(C) such that A is similar to B but not congruent. Find 2 matrices A and B in M2(C) such that A is congruent to B but not similar. What is the best strategy to find such…
arm46
  • 343
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Why is the transpose related to the dual space?

A matrix with $m$ rows and $n$ columns in the real numbers is a map from $M : \mathbb{R}^n \to \mathbb{R}^m$; the transpose of this matrix is then a map $M^T: \mathbb{R}^m \to \mathbb{R}^n$. However, it seems like the transpose is related to the…
greg115
  • 443
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Prove that if AB-BA=B then A and B share a common eigenvector

To clarify, this is a question from a contest--It's from a Chinese contest, a problem from year 2010. $A$ and $B$ are operators $V\to V$ on the complex vector space $V$. I've attempted the problem but seem to have come to some contradictions:…
LGu
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Suppose $T$ is an operator on $V$ and $T^2 = I$ and $-1$ is not an eigenvalue of $T$. Prove that $T = I$.

My attempt: Since $ -1$ is not an eigenvalue of $T\implies T+I$ is invertible, that is $(T+I)^{-1}$ exists. Now $(T+I)(T-I) = T^2 - I \implies (T+I)(T-I) = 0$. After applying $(T+I)^{-1 }$ on left and right hand side of the previous equation we…
Darby Bond
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Is $\{\sin x,\cos x\}$ independent?

Is $\{\sin x,\cos x\}$ linearly independent in $\mathbb{R}^n$? I thought they were not because I can write $\cos x=\sin (x+\pi/2)$. My professor on the other hand said it was independent and his proof is as follows: If $\{\sin x,\cos\}$ is…
Sujaan Kunalan
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Can there be "polynomial spaces"?

I don't know how to better frame this question. Thinking about vector spaces and their role in basically everything Calculus touched, I can understand why they are so central, especially in areas like differential geometry. But Taylor's Theorem got…
Lourenco Entrudo
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Understanding Dual Transformations and reasoning behind definition

In Linear Algebra working with Dual space and dual transformations I've come along this very basic definition of the dual transformations: Suppose: $T^*$ is a dual transformations from $W^*\to V^*$ $T$ is a linear transformation from $V\to W$ …
vondip
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Orthogonal vectors in complex vector space

Consider $u,v\in \mathbb{R}^2$ where $u=(2,1), v=(-1, 2)$. $u$ and $v$ are orthogonal since $u\cdot v=0$. If we put them in $\mathbb{C}$, they should be still orthogonal. However, $ \langle u,v \rangle=(2+i)(-1+2i)\ne 0$. I must misunderstand…
Precalculus
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Proving there is no squared $n\times n$ (n odd) real matrix yielding minus the identity matrix

Prove or give a counterexample There is no $A \in \Bbb R^{3 \times 3}$ such that $A^2 = -\Bbb I_3$ Here's my attempt I suspect that the statement is true. To prove it, I used contradiction Assuming $A^2 = -\Bbb I_3$ holds, we take the determinant…
JD_PM
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