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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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A sufficient condition for diagonalization

How to prove that if $A,B,M\in \mathcal{M_n}(\mathbb{C})$ and $\lambda,\mu\in\mathbb{C}$ so $$ \begin{cases} M&=& \lambda A+\mu B \\ M^2&=& \lambda^2 A+\mu^2 B \\ M^3&=& \lambda^3 A +\mu^3 B \end{cases} \Rightarrow M\ \text{is…
user63181
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If $TS=ST$, then $S=\alpha T+\beta$.

Let $T=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ be a non-scalar matrix. If $S=\begin{pmatrix}e&f\\g&h\end{pmatrix}$ be such that $TS=ST$. Why there exists $\alpha,\beta\in \mathbb{C}$ such that $$S=\alpha T+\beta I\;?$$ Note that $TS-ST=0$ is…
Schüler
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What inspires people to define linear maps?

I am currently self studying Linear Algebra Done Right. I am doing OK and currently on Chapter 3 Linear Maps. My understanding now is that a map is like a function that maps something to another thing. Why people like to particularly define linear…
JOHN
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Invariant subspaces

Good morning, Let T be a linear transformation, which acts on a vector space V, over a field F. Let W be a sub-space which invariants of T, and f,g polynomials from the same field F. I need to prove that W is invariant of g(T) and f(T)(W)…
user6163
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let $A\in M{_n}\ (\mathbb R)$ s.t $A^2=I$ such that $A\neq I$, $A\neq -I$ how prove $-(n-1)\le tr A\le n-1$

let $A\in M{_n}\ (\mathbb R)$ s.t $A^2=I$ such that $A\neq I$, $A\neq -I$ how prove $-(n-1)\le tr A\le n-1.$ Thanks in advance
M.H
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Existence of a solution with entries $1,-1$ or $0$

Suppose $A$ is an $m\times n$ matrix and $b$ is an $m\times 1$ vector (where $m,n\geq 3$) such that each of the column vectors of $A$ and the vector $b$ has one entry equal to $1$, another entry equal to $-1$ and rest of the entries are zero. Now if…
pritam
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Question on a proof about the Rank of a Matrix

The question is: Give a formal proof for the following statement: Given a matrix A and a scalar c, show that rank(cA) = rank(A) Here are the steps that I took to go about the proof: (1) Prove this claim: Let v1, v2, ..., vN be vectors then {v1,…
Cecile
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Why do similar matrices represent the same linear transformation?

I don’t understand the theorem: $A$ and $B$ are similar if and only if they represent the same linear transformation. I know one direction "If $A$ and $B$ represent the same linear transformation then they are similar", but why is the other…
whwjddnjs
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Composition of orthogonal projections

I need to prove the following result: Suppose $P_1$ and $P_2$ are orthogonal projections onto closed subspaces $V_1$ and $V_2$, then $P_1P_2x = x$ if and only if $x\in V_1\cap V_2$. But it seems to me that if you take two parallel lines $V_1$, $V_2$…
user61408
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Question on finite dimensional vector space

Let $V$ be a finite dimensional vector space over $F$, a finite field of two elements. Is it possible to find the sum $$\sum_{v\in V}v$$?
gtolessa
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If $A$ is a square matrix that satisfies $A^2-A+2I=0$, show that $A+I$ is invertible

If $A$ is a square matrix that satisfies $A^2-A+2I=0$, show that $A+I$ is invertible. I understand how to find if $A$ is invertible but I don't know how to solve for the $A+I$ version.
user612325
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Let $P_3$ be the space of polynomials of degree $\leq 3$ . Find the kernel and the image of the linear map $f(x) \mapsto f(x + 1)−f(x)$

Let $P_3$ be the space of polynomials of degree $\leq 3$ over the field $\mathbb{Z}/2\mathbb{Z}$. Find the kernel and the image (that is, give bases of these spaces) of the linear map $f(x) \mapsto f(x + 1)−f(x)$ So for this problem, a basis for…
depaul
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Showing that matrix is invertible using eigenvalues

Let $A$ be matrix from the vector space of square $N \times N$ matrices. With the inital information: $A^2-4A=4I$. How does one show that $A+I$ is invertible? (I need please a solution that involves eigenvalues) Thank you
user6163
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A question about a method that shows $\mathbb{R} $ not finite dimensional.

Upon looking at methods that show $\mathbb{R}$ is not finite dimensional over $\mathbb{Q}$ I came across a method mentioned here by the user Bill Dubuque, he took a set of vectors of the form $\log(p)$ where $p$ is prime and showed that the set is…
user10444
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Trace of a matrix times outer product

$\DeclareMathOperator{\trace}{tr}$Is there any relationship between $\trace(Sxx^T)$ and $x^TSx$? Is there a nice way to write the set of quadratic functions of the components of a vector $x$ given coefficients in some matrix $S$?
Neil G
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