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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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An inequality for the dimension of the sum of subspaces

The answer with the most of upvotes on MO is this answer on $\dim(U+V+W)$. Question: 1. Is it nonetheless true that every three vector subspaces $U$, $V$ and $W$ of a vector space $M$ satisfy $$ \dim(U +V + W) \le $$ $$ \dim U + \dim V + \dim W -…
Sebastien Palcoux
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A question about the odd sized minors of a certain matrix

In Dimers and Amoebae by Kenyon, Okounkov, and Sheffield (2003), they say that it is easy to see that for matrices of the form $$ \left( \begin{array}{ccccc} a_1 & 0 & 0 & 0 & b_n \\ b_1 & a_2 & 0 & 0 & 0 \\ 0 & b_2 & a_3 & 0 & 0 \\ 0 & 0 & b_3 &…
Pjotr5
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Showing Unit sphere is convex

Good evening guys! I have to show that the unit sphere represented by is convex. A set is said to be convex when $sx + (1 - s)y \in M$, where $x, y \in M$ and $s \in (0,1)$ I've read on wikipedia that this can be proven over the triangle…
Clash
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$T^2=I$ implies $T$ is diagonalizable

Suppose $T:V\rightarrow V$ is linear and $T^2=I$. Prove that $T$ is diagonalizable. First, I know that $T$ has only eigenvalues 1 or -1. Also I observed that $(T-I)(T+I)=0$, does this fact help to show that $T$ is diagonalizable?
user112358
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Uniqueness of Unitary Similarity Transform

Let's say I have a Hermitian matrix $H$ that is diagonalized by a unitary matrix $U$: $D = U^{\dagger} H U$, where $D$ is diagonal. How unique is $U$? If I stick on an overall phase factor by $U \rightarrow e^{i\phi}U$, then $U$ still diagonalizes…
kokopelli
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A bad Cayley–Hamilton theorem proof

Given $A\in M_{n \times n}(\mathbb{F})$ and $p_{A}(x)=\det(xI-A)$ why saying that $\det(AI-A)=0$ is not valid?
Yinon Eliraz
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Geometric meaning of outer product of a vector with itself

This question is related to the question in the link below: Is there a geometric meaning to the outer product of two vectors? The answer is clear, but I am wondering: If we take a outer product of a vector with itself, then is there a specific…
Creator
  • 3,128
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Does anyone know an interesting introductory topic involving vector spaces over the rationals

Many introductory books on vector spaces mention that the scalars need not be reals, and might even have sections discussing complex vector spaces or vector spaces over the integers mod 2. I have never seen any such book mention that all of the…
Barry Smith
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2 answers

Linear Algebra: System of Equations

Consider a finite sequence $x_i \in (0,1)$ for $i=1,\ldots, n$ and define $y_i=\dfrac{\Pi_{j=1}^n x_j }{x_i}$. I solved this system for $x$ in terms of $y$ and got $$x_i=\dfrac{\left(\Pi_{j=1}^n y_j \right)^\frac{1}{n-1}}{y_i}.$$ Now pick some…
Sergio Parreiras
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Dimension of commutator space of matrices (again)

For a square matrix $A\in F^{n\times n}$ over a field $F$, define the commutator subspace $C_A = \{ B\in F^{n\times n} \vert AB = BA\}$ of matrices which commute with $A$. This other question by RiaD asks for a proof that $\dim C_A\geq n$ for any…
Noah Stein
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Dimension of set of commutable matrices

Let $A$ be an $n \times n$ complex matrix and $V = \{B\mid AB=BA\}$. I've proved that $V$ is a vector space. How can I prove that $\dim V \ge n$ for any $A$?
RiaD
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How does one prove that if $f$ and $g$ are linear functionals on $V$ such that $h=fg$ is also a linear functional, then either $f=0$ or $g=0?$

I am self-studying Hoffman and Kunze's book Linear Algebra. This is the exercise 13 from page 106. Let $\mathbb{F}$ be a subfield of the field of complex numbers and let $V$ be any vector space over $\mathbb{F}.$ Suppose that $f$ and $g$ are linear…
user23505
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1 answer

What are these math symbols?

I'm studying linear algebra and all of a sudden the symbol $\dot{+}$ appears. For example: $a*(v \dot{+} w) = a*v \dot{+} a*w$ Any idea what it might be? Also two more symbols. they are on top of $0$ in equations: - and ~. For example $u \dot{+} w…
RedRose23
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If two Normal operators commute then the product is normal

Let $T$ and $S$ be two normal operators in a infinite dimensional inner-product complex vector space. If $ST=TS$, I want to show that $TS$ is normal. For the finite-dimensional case, it went down to showing $T^*=f(T)$ for some polynomial $f$. But I…
Jonas Gomes
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Multiplication of inverse and non-inverse matrices

I have thought about the combination of multiplication product of invertible and non-invertible matrices: invertible $\cdot$ invertible = invertible non-invertible $\cdot$non-invertible = non-invertible non-invertible $\cdot$invertible =…
gbox
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