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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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Which one about the rank and nullity is true

Let $A$ and $B$ be $n$ $\times$ $n$ real matrices such that $AB$=$BA$=$0$ and $A+B$ is invertible. Which of the following are always true? $1$. rank $(A)$=rank $(B)$. $2$. rank $(A)$$+$rank $(B)$ =$n$ $3$. nullity $(A)$ $+$ nullity $(B)$ = $n$ $4$.…
Topology
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An inequality on a matrix involving a norm and the eigenvalues

Let $A = \left( {a_{ij} } \right)$ be a matrix over $\mathbb R$, of size $n \times n$. Let $\left\{ \lambda _k \right\}_{k = 1}^n$ be the $n$ eigenvalues of the matrix. Prove the following inequality: $$ \sum_{k = 1}^n \left| {\lambda _k }…
August
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Quadratic form under $\mathbb{C}$ finite-dimension vector space always have a nonzero vector that yields zero?

Let $ V $ be a vector space of finite dimension on $\mathbb{C} $ and $ \dim(V) \gt 1 $. Show that for every quadratic form $ q : V \to \mathbb{C} $ there exists $ 0 \neq v \in V $ such that $ q(v) = 0 $. It certainly has something to do with the…
dan
  • 81
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Matrix invertability clarification and understanding

I am trying to set up some "mental models" for how to think about matrix invertability. I am currently studying linear algebra on a basic level and I would please like some explanations to the question below, general information about matrix…
Lukas Arvidsson
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4
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6 answers

Kernel of linear transformation in $\Bbb R^3$

Let $T: \Bbb R^3 \rightarrow \Bbb R^3$ be a linear transformation satisfying \begin{align*} T(0,1,1) =& (-1,1,1) \\ T(1,0,1) =& (1,-1,1) \\ T(1,1,0) =& (1,-1,0) . \end{align*} Is it necessary true that $\ker(T) = \operatorname{Sp}\{(1,-1,1)\}$…
Ilan Aizelman WS
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4
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Do SVD and Jordan imply that any operator can be represented through diagonal/almost diagonal matrix?

Good morning. Assume that we are given a complex vector space. I know that SVD guarantees a factorization of form $M = USV^{*},$ and the Jordan form gives us a factorization of form $M = PJP^{-1}.$ My question is the following: does this mean that…
Matvey
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Is there a symbol for the number of dimensions in a vector?

Here's an equation from a text book for computing a unit vector: $$\hat v = \frac{\overline v}{ \sqrt{ \sum ^n _{i=1} (\overline v_i)^2 } }$$ Now I may be wrong here, but using $n$ doesn't really cut it here for me. $n$ can't be any old amount, it…
Starkers
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3 answers

Can an abelian group be turned into two nonisomorphic vector spaces by different actions of the same field?

I was wondering if you could find an abelian group and define two different actions of a field on it such that the resulting vector spaces are of different dimensions over the field. I tried various extensions of $\mathbb Q$. Now, since the set $V$…
Nishant
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1 answer

The exponential of a real matrix

Let A be a $n \times n$ real matrix such that $Exp(A) \in SO(n)$, is it necessarily that $A$ is anti-symmetric?
Summer
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4
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2 answers

Show $A$ hermitian $\iff v^tA\overline{v}\in\Bbb R$ for all $v \in \mathbb{C}^n$

Show that a matrix $A \in M(n \times n, \mathbb{C})$ is hermitian iff $v^tA\overline{v} \in \mathbb{R}$ for all $v \in \mathbb{C}^n$.
user149868
4
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2 answers

Proving that two right and left eigenvectors are not orthogonal.

Let $A$ be a square non-Hermitian matrix and $c$ be an eigenvalue of $A$ with algebraic multiplicity $1$. Let $Ax = cx$ and $y^{H}A = cy^{H}$ where $y^{H}$ is a conjugate transpose of $y$. Prove that $y^{H}x \neq 0$. Please give me a hint. Thanks.
fiverules
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5 answers

Any trick on finding the inverse of this matrix?

Supposing I have a matrix, $\pmatrix{0&0&\lambda\\0&\lambda&-1\\ \lambda&-1&0}$. Without question you can work out the inverse if this matrix. But since it is highly structured, I suppose there should be some quick way to find out the inverse of…
Jack2019
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Confusion about row reduction and linear independence of a column vector.

I am quite new to studying linear algebra and I am struggling with the concept of writing a set of vectors in column form and finding out if the set of vectors are linear independent (I feel like I might have missed something fundamental). So here…
Lukas Arvidsson
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4
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1 answer

do two vector spaces have the same linear dimension?

Let $Z$ be a countable set. Let $f_1,....,f_n$ be a collection of real functions over $Z$. Let $z_1,...,z_m,...$ be an enumeration of elements of $Z$. Define $V_1 = \{ (f_1(z),...,f_n(z)) | z \in Z\}$ a set of $n$ dimensional vectors. Define $V_2…
normvector
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0 answers

Are these vectors a base of the Subspace?

$a=(1,2,3,0)$ $b=(0,3,2,4)$ $a$ and $b$ form the subspace $U$ is the base $B(a,b)$ a base of $U$? My guess is yes, because they are linearly indpendent?
SuperNova
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