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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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Multiplicity of eigenvalues

Suppose $A$ is an $n\times n$ complex matrix. How to show the following two properties If $\lambda$ is an eigenvalue of $A\bar{A}$, so is $\bar{\lambda}$. Here $\bar{A}$ means the entrywise conjugate of $A$. The algebraic multiplicity of negative…
Sunni
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An algebra of nilpotent linear transformations is triangularizable

How to prove "An algebra of nilpotent linear transformations is triangularizable" using linear algebra only?
Sunni
  • 4,536
8
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2 answers

dimension of intersection of hyperplanes

What are the possible dimensions of intersection of k-number of hyperplanes in $\mathbb{R}^n$ ? I look at some examples in lower dimension but I cannot come with a nice cases according to which the dimension varies. Thanks for your valuable time.
GA316
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8
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the determinant function is an open function?

The determinant function $\det:M(n,\mathbb R)\rightarrow \mathbb R$ is an open mapping or a closed mapping? The determinant function $\det:M(n,\mathbb C)\rightarrow \mathbb C$ is an open mapping or a closed mapping?
David Chan
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Matrix having to be orthogonal, knowing it's norm-preserving

If we know that a real $m \times m$ matrix $C$ is norm-preserving, $||C\textbf{v}|| = ||\textbf{v}||$, then $C$ has to be orthogonal. Why should this be the case?
MathsWanderer
  • 81
8
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Prove that A(AB-BA) = (AB-BA)A implies AB-BA is nilpotent.

Let A and B be $n \times n$ complex matrices such that $A(AB-BA) = (AB-BA)A$ a) Show that for every positive integer $k$, the matrix $(AB-BA)^k$ is of the form $AC-CA$, where $C$ is an $n \times n$ complex matrix. b) Prove that $AB-BA$ is…
rackne
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2 answers

Is every family of almost commuting matrices close to some family of commuting matrices?

Suppose that there are many matrices $A_i\in M_n(C)$, $i=1,2,3,\cdots,m$, that almost commute with each other (what I means is that they are not commute with each other, but $||[A_i, A_j]||$ is very small). Can I slightly change the matrices, e.g.,…
D. J. Zhang
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Angle between two 4D vectors

I was wondering if there is any difference between finding the angle between two 4D vectors as opposed to finding the angle between two 3D vectors? I have $u = (1, 0, 1, 0)$ , $v = (-3, -3, -3, -3)$ and used the dot product to find the angle between…
jn025
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How to define sparseness of a vector?

I would like to construct a measure to calculate the sparseness of a vector of length $k$. Let $X = [x_i]$ be a vector of length $k$ such that there exist an $x_i \neq 0$ . Assume $x_i \geq 0$ for all $i$. One such measure I came across is defined…
Learner
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Prove that Either $T$ Is Diagonalizable or $T$ Is Nilpotent.

(Linear Algebra - Hoffman, Kunze, 2nd Ed., Sec 6.8, Q6) Let $V$ be a finite-dimensional vector space over the field $\mathbb{F}$, and let $T$ be a linear operator on $V$ such that $\textrm{rank} (T) = 1$. Prove that either $T$ is diagonalizable or…
Abdul3333
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Analogy of transpose for a function?

In the page 2 of Linear algebra explained in four pages reference, it has a box describing the relationship between functions and linear transformation. It states that the set of zeroes of a function ($f(x)=0$) is analogous to the null space of a…
Ming-Tang
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How do I prove that a matrix is a rotation-matrix?

I have to prove that this matrix is a rotation-matrix $$\begin{pmatrix} \frac12 & 0 & \frac{\sqrt{3}}{2} \\ 0 & 1 & 0 \\ \frac{\sqrt{3}}{2} & 0 & \frac12 \end{pmatrix}$$ How do I do this? My idea is to multiplicate it with $\begin{pmatrix} x \\ y…
Christian
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4 answers

Linear Algebra solution when determinant is zero

I am doing practice questions in my book and I came upon this True/False question: If $\det(A) = 0$, then the linear system $Ax=b$, $b\neq 0$, has no solution. The book is saying that the answer is false. But why is that? I thought the answer is…
user2989964
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Is the limit of the dual spaces the dual space of the limit?

Note: Whenever I say "dual space" in this question, I mean the algebraic dual space. Consider an infinite-dimensional vector space $V$. Since it is infinite-dimensional, its double-dual $V^{**}$ is strictly larger than $V$. On the other hand,…
celtschk
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Matrix solution to $X-AXA=K$

This came up in a practical problem involving a state change in a digital filter system. Find $X$ given $A$ and $K$, where $A,X,K$ are all $n$ x $n$ square matrices: $$ X - A X A = K $$ I couldn't find a direct way to do this, eventually I…
greggo
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