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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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Can eigenvalues be interpreted as coordinates in a vector space of projection operators?

Consider the vector space $\mathbb R^n$. Let $T : \mathbb R^n \to \mathbb R^n$ be a linear operator, and let $\mathbf T$ be its matrix representation with respect to the standard basis in $\mathbb R^n$. Suppose that the eigendecomposition of…
mhdadk
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Artin's Algebra 4.4.8

Q. Let $T$ be a linear operator on a finite-dimensional vector space for which every nonzero vector is an eigenvector. Prove that $T$ is multiplication by a scalar. I did not find this question in old posts. Approach:- First suppose…
Sonu
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If $A$ is a $3\times 3$ complex matrix such that $A^3=-I$, then $A$ has distinct eigenvalues?

Let $A$ be a $3\times 3$ complex matrix such that $A^3=-I$ How to show that $A$ has distinct eigenvalues? What if i consider $A=-I?$ Isn't then -1 becoming the only eigen value?
ddabir
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Casimir element invariance

I'm sorry to bother but i'm having some problems in proving that, given a simple Lie Algebra L of finite dimension $n$ (equipped with the Killing form) and its enveloping universal algebra U(L), then the element (Casimir): c = $\sum x_iy_i$ where…
claudia
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Every subspace of $\mathbb{R}^n$ is a solution space of a homogeneous system of linear equation.

All solution of $AX = 0$ where $A$ is a $n \times n$ matrix and $X$ is a column vector form a subspace of $\mathbb{R}^n$. All the subspaces of $\mathbb{R}^n$ are of this type. How to prove this result? Linear Algebra: solution of homogeneous system…
Supriyo
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row echelon vs reduced row echelon form

I apologize if this is a very basic question. I understand the difference between the two forms, but i was curious when row echelon from is enough. where is row echelon form used?. Why shouldn't I always go for reduced row echelon form?
Surya
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Given a matrix $A$ and what it maps two vectors to, is $0$ an eigenvalue of it?

Studying for my Algebra exam, and this question popped out with no solution in a previous exam: Given a matrix $A$ such that $A \begin{pmatrix} 0 \\ -1 \\ 0 \end{pmatrix} = \begin{pmatrix} -2 \\ -4 \\ 6 \end{pmatrix},\ A \begin{pmatrix} 2 \\ 0 \\ 0…
TheNotMe
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Why is a linear system of equations considered consistent when a solution contains 0/0?

Suppose I have a system of equations as such, with $\alpha$ and $\beta$ being constants: $$ x_1+5x_2+x_3=1\\ x_1+6x_2-x_3=1\\ 2x_1+\alpha x_2-6x_3=\beta $$ Suppose I want to calculate for which values of $\alpha$ and $\beta$ my system is consistent.…
Elijah
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what is degree of minimal polynomial?

Let $V$ and $ W$ be finite dimensional vector space over $\mathbb R $ and let $T_1 : V \rightarrow V$ and $T_2 : W \rightarrow W$ be linear transformation whose minimal polynomial are $f_1 (x)= x^3+x^2+x+1$ and$f_2 (x)= x^4 - x^2-2$. let $T : …
user45799
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This matrix defined by a polynomial $f$ has rank less than $\deg(f)+1$

Let $f\in \mathbb{R}[x]$ be a polynomial of degree $d$. Let $x_1, \dots, x_n$ be real numbers, I want to show the matrix $A$ given by $A_{ij} = f(x_i + x_j)$ has rank $\le d+1$. My attempt: I tried writing the polynomial as $f(t)=\sum_{k=0}^d c_k…
RunningMeatball
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Faster method for solving the inequality $\sqrt{λ_1^2+ λ_2^2+ λ_3^2}$ $\leq$ $\sqrt{1949}$ for a 3x3 matrix

λ1, λ2, λ3 are the eigen values of the matrix \begin{bmatrix}26&-2&2\\2&21&4\\4&2&28\end{bmatrix} Show that $\sqrt{λ_1^2+ λ_2^2+ λ_3^2}$ $\leq$ $\sqrt{1949}$ Here's my approach at solving this problem - Performing Single Value Decomposition of…
ksuyft
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Condition on endomorphisms satisfying $\operatorname{Ker} f=\operatorname{Im} f$

Let $E$ a finite dimensional vector space and $f \in \mathcal{L}(E)$. Show that: $$\operatorname{Ker} f=\operatorname{Im} f \Leftrightarrow (f^2=0) \wedge (\exists h \in \mathcal{L}(E) \mid h \circ f + f \circ h=Id_E)$$ My attempt: I managed to…
Jujustum
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The form of a solution in a linear system

I have this linear system: $\left\{\begin{array}{c} 2x + 3y - 4z = \ 1 \\ 3x - y - 2z = 2 \\ x - 7y - 6z = 0 \end{array}\right.$ I found the following solution: $\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \alpha…
Katy23
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Finding a unitary matrix that diagonalizes a given matrix

Let $$T=\begin{pmatrix}5 & 0 & 0 \\ 0 & 2 & i\\ 0 & -i & 2 \end{pmatrix}$$ be a Hermitian matrix. I found the eigenvalues and eigenvectors already and they are $1,3,5$ and $\begin{pmatrix}0\\-i\\1\end{pmatrix}$,$\begin{pmatrix}0\\i\\1\end{pmatrix}$,…
emka
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Confirmation of correctness in proof regarding norm preserving operator

I just want to know if my solutions are correct for the following problem (euclidean norm assumed): A linear transformation $T:\Bbb R^n \to \Bbb R^n$ is norm preserving if $|T(x)| = |x|$ for all $x \in \Bbb R^n$, and inner product preserving if…
Gold
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