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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.

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Let $\alpha$ be a solution of a monic quadratic polynomial with integer entries and $|\alpha|=1$.Then prove that $\alpha^{12}=1$.

($a$) Let $\alpha$ be a solution of a monic quadratic polynomial with integer entries and $|\alpha|=1$.Then prove that $\alpha^{12}=1$. ($b$)Let $A \in M_2(\mathbb{Z})$ such that $A^n=I$ for some $n$ then show that $A^{12}=I$ completely stuck on…
amiow
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Identities with Adjoints

The classical adjoint $\operatorname{adj}(A)$ of a square matrix $A$ has its $(i,j)$-th entry equal to the $(j,i)$-th cofactor (signed minor) of $A$. If $\det(A)\neq0$ we can define the inverse $A^*$ of $A$ as $\operatorname{adj}(A)/\det(A)$ and use…
Christopher Sokolnicki
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$A^3+A=0$ We need to show $\mathrm{rank}(A)=2$

Let $A\ne 0$ be a $3\times 3$ matrix with real entries such that $A^3+A=0$. We need to show $\mathrm{rank}(A)=2$. $\det A(A^2+I)=0\Rightarrow\det A=0\Rightarrow \mathrm{rank}(A)<3$, Suppose $\mathrm{rank}(A)=1$, Then I showed one matrix with rank…
Myshkin
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Orthogonal linear transformations and the dot product

I am trying to solve a problem on the dot product, and I do some manipulations and come to the conclusion that $\langle x, A^{T} Ax \rangle = \langle x,x \rangle$. $x$ is a column vector with $n$ rows, and $A$ an $n \times n$ matrix. I know in…
user38268
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How can I show if $AA^T+A^TA=A+A^T$, then $Col(A)=Col(A^T)$?

Question: For a square matrix $A$, if $$AA^T+A^TA=A+A^T,$$ prove that $$\text{Col}(A)=\text{Col}(A^T).$$ I want to prove this statement, but it's quite difficult... I think I should use "$Null(A)$ is orthogonal to $Col(A^T)$" Any good ideas?
Steve Kim
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Annihilator of a subspace

Let $U$ and $V$ be vector spaces over a field $F$, and let $T: U\rightarrow V$ be a linear transformation, with $T^*: V^*\rightarrow U^*$ the corresponding adjoint. We would like to show that $\text{Im}(T^*)=\text{Ker}(T)^{\perp}$, where $S^{\perp}$…
yoshi
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Please correct my work, finding Eigenvector

Determine whether matrix A is a Diagonalizable. if it is , determine matrix P that Diagnolizes it and compute $P^{-1}AP$. $$A= \begin{bmatrix} +3 & +2\\ -2 & -3\\ \end{bmatrix} $$ $$A-\ell I = …
Node.JS
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Is there a linearly independent spanning set for $\Bbb{R}$ with respect to $\Bbb{Z}$?

Is there a set $S\subset \Bbb{R}$ such that every $x\in\Bbb{R}$ can be writen as $$x = a_1 s_1 + a_2 s_2 + \dots + a_n s_n$$ where $a_1, a_2, \dots, a_n\in\Bbb{Z}$ and $s_1, s_2, \dots, s_n\in S$, and $x = 0 \implies a_1 = a_2 = \dots = a_n =…
Alma Arjuna
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Finding all planes that satisfy two conditions

Let $\Pi_1 : 7x-5y-2z=0, \Pi_2 : 5x-4y-z=0$ and $\mathbb{L}$ the line that passes through points $P=(-2,3,-3)$ and $Q=(-1,2,-1)$. Find all planes $\Pi$ that satisfy: $\Pi \cap \Pi_1 \cap \Pi_2 = \emptyset$ $d(R,\Pi)=\sqrt{14} \forall R \in…
Fernando Martin
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Does the dim[rowspace] ALWAYS equal dim[columnspace]?

My professor was hinting this was going to be on the exam, but wasn't telling us if this is true. I do believe in fact it is true though, because both the rowspace and column space are determined by the number of leading 1's in the row-reduced…
Johnathon Svenkat
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Number of Invariant Subspaces of a Jordan Block

I'm asking this question on behalf of a person I'm supposed to be tutoring who has this problem as part of eir homework. The problem is "How many invariant subspaces are there of a transformation $T$ that sends $v\mapsto J_{\lambda,n}v$" where…
Eric Stucky
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Calculating $n$-th power of a matrix

I was doing an exercise of a past exam in which one of the things I had to do was calculating the $n$th power of a Jordan matrix $$J=\begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{pmatrix}.$$ I started calculating until the 5th power but I…
Alberto De Celis Romero
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Is it true that ${\rm Tr}((A^{1/2}BA^{1/2})^{1/2}) = {\rm Tr}((BA)^{1/2})$ for positive semidefinite matrices $A,B$?

If $A,B$ are positive semidefinite matrices, then prove or disprove the following, $${\rm Tr}((A^{1/2}BA^{1/2})^{1/2}) = {\rm Tr}((BA)^{1/2})$$ I verified numerically in MATLAB, and apparently the following seems to be true (at least for some…
user550103
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The Linear Independence of {$\sin(2^m x), \sin(2^{m-1}x), \ldots, \sin(2x), \sin(x)$}.

Let $m \in \mathbb{N}$. I wish to prove the linear independence of the set of vectors {$\sin(2^m x), \sin(2^{m-1}x), \ldots, \sin(2x), \sin(x)$} $\subset F(\mathbb{R})$. I had attempted to prove the property by induction upon $m$: the base case,…
V. Elizabeth
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All nondegenerate bilinear symmetric forms on a complex vector space are isomorphic

All nondegenerate bilinear symmetric forms on a complex vector space are isomorphic. Does this mean that given a nondegenerate bilinear symmetric forms on a complex vector space that you can choose a basis for the vector space such that the matrix…
user637978
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