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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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Generalized Eigenvalue Problem

Consider a generalized Eigenvalue problem $Av = \lambda Bv$ where $A$ and $B$ are square matrices of the same dimension. It is known that $A$ is positive semidefinite, and that $B$ is diagonal with positive entries. It is clear that the generalized…
John
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Derivative of Determinant Map

For $ V= ( V_1, V_2) $ and $ W= ( W_1, W_2) $, given a determinant map $ \det : \mathbb{R}^2\times \mathbb{R}^2\rightarrow \mathbb{R}$ defined as $ \det (V,W)= V_1W_2-V_2W_1$. Then have to find the derivative of the determinant map at $( V, W)\in…
preeti
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Jordan decomposition of $A^T$ given that of $A$

Suppose I have the Jordan normal form of a matrix $A$. The decomposition involves the Jordan matrix $J$ and a similarity matrix $P$ such that $P^{-1}.J.P = A$. My question: is it possible to find the similarity matrix of $A^T$ given that we know…
Bhuvanesh
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(Homework) Prove that there is no T-invariant subspace $W_2$ such that $R^2$ = $W_1 \oplus W_2$

Let T be a linear operator on $R^2$ defined by T(x,y) = (2x+y, 2y) and $W_1 = span{(1,0)}$ Prove that there is no T-invariant subspace $W_2$ such that $R^2 = W_1 \oplus W_2$ At first I showed that $W_1$ is an eigenspace and T is not diagonalizable…
Sai
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Change of basis and inner product in non-orthogonal basis

I have a vector, originally expressed in the standard coordinates system, and want to perform a change of basis and find coordinates in another basis, this basis being non-orthogonal. Let $B = \{e_1, e_2\}$ be the standard basis for $\Bbb R^2$. Let…
joseph M'Bimbi-Bene
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Linear combination of a vector and its negative

I'm having trouble understanding the question's answer from #19. How does the combination of a vector v and its negative fill a half space? Doesn't it only fill a line?
John
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Rule for squaring arbitrary powers?

This is a really simple question, but I don't know how to phrase it well enough for Google. I'm going through a proof and don't understand how: $$ (q^{2^{n+1}})^2 = q^{2^{n+2}} $$ I thought it would be $q^{4^{n+1}}$instead, or are they equivalent?…
hohner
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Adjoint and Adjugate are same or different?

The notions of adjoint and adjugate, which I saw, are as follows: (1) Let $T:V\rightarrow W$ be a linear map. Then there is a corresponding linear map between the duals of these spaces: $T^*:W^*\rightarrow V^*$, defined as follows: for every linear…
Groups
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Show that four points are coplanar

I read all posts online regarding how to show four points are coplanar. However, none of them discuss the idea behind the method. Can someone explain how the triple scalar product works?
LKSR
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Understanding a proof of corollary of Farkas lemma

I'm trying to understand a proof of a corollary to the Farkas lemma in some lecture notes. For completeness sake I'll first state the Farkas lemma and then the corollary + proof, as stated in these lecture notes. (Farkas lemma) Given $A \in…
Stijn
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Prove that congruent matrices have the same rank.

Can someone prove that two similar matrices have the same rank? Thanks a lot.
bob
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Product of reduced row-echelon matrices is also reduced row-echelon

Show that the product of two reduced row-echelon matrices is also reduced row-echelon. That's what I think: A reduced row-echelon matrix has columns like $e_1 =(1, 0, \cdots , 0)^T$ and $e_2 =(0, 1, 0, \cdots , 0)^T$. For columns in between…
Nighty
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$A$ is diagonalizable if $A^8+A^2=I$

Given a matrix $A\in M_{n}(\mathbb{C})$ such that $A^8+A^2=I$, prove that $A$ is diagonalizable. So let $p(x)=x^8+x^2-1$ and we know that $p(A)=0$. The next step would be to show that the algebric and geometric multipliciteis of all the eigenvalues…
Yinon Eliraz
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Are a row vector and a column vector the same thing?

Suppose I have $$ A = \begin{bmatrix} a\\b\\c\\d\\ \end{bmatrix}$$ $$ B = \begin{bmatrix} a& b& c & d\\ \end{bmatrix}$$ Now, I know $A = B^T$. But in what sense are these different mathematical objects?…
Stan Shunpike
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If $\alpha_{1},\ldots,\alpha_{m}$ are vectors different from zero vector, then there is a linear functional $f$ such that $f(\alpha_{i})\neq 0$

I am self-studying Hoffman and Kunze's book Linear Algebra. This is exercise 14 from page 106. Let $\mathbb{F}$ be a field of characteristic zero and let $V$ be a finite-dimensional vector space over $\mathbb{F}$. If …
user23505
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