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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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If the sum $A_1 + A_2 + · · · + A_n$ is equal to the negative of the identity operator on $V$ , show that $dim_{\mathbb{R}} V$ is even.

Let $V$ be a finite-dimensional vector space over the real numbers $\mathbb{R}$. Suppose $A_1,A_2,....,A_n$ are finitely many pairwise commuting linear operators on $V$. Assume that none of the operators $A_i$ has a negative real eigenvalue. If the…
user348342
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Application of rank of a matrix

Are there any real life applications of the rank of a matrix? It need to have a real impact which motivates students why they should learn about rank.
matqkks
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What is the meaning of subtracting from the identity matrix?

If I subtract the matrix $A$ from the identity matrix $I$, $I - A$, is there a meaning to the resulting matrix perhaps given some conditions like invertibility or symmetry? For example, $$ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} -…
Chewers Jingoist
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Does a scalar represent a linear map?

I have been wondering recently if a $1 \times 1$ matrix reprsents a scalar, and after doing some reading I'm still not satisfied. I've decided to ask this question from a different perspective: does a scalar represent a $1 \times 1$ map? Because any…
user3210986
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Irreducible minimal polynomial implies every invariant subspace has an invariant complement

Full version of the problem is following: Let T be a linear transformation on a finite dimensional vector space $V$ over a field $\mathbb{F}$. If the minimal polynomial $p_t$ of T is irreducible, then every T invariant subspace $W$ has a…
Yeonjoo Yoo
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Complex matrix that commutes with another complex matrix.

I am trying to learn some linear algebra, and currently I am having a difficulty time grasping some of the concepts. I have this problem I found that I have no idea how to start. Assume that $\bf A$ is an $n\times n$ complex matrix which has a…
Melky
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How to deal with $A^{26}=I$?

I got stuck in this problem: Let $A:\mathbb{R}^{6}\rightarrow \mathbb{R}^{6}$ be a linear transformation. Assume $A^{26}=I$, prove that $R^{6}=\oplus_{i=1}^{3} V_{i}$, with $AV_{i}\subset V_{i}$(the explicit condition is $V_{i}$ are 2-dimensional…
Bombyx mori
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For every nonzero vector $v$ there exists a linear functional $f$, sucht that $f(v) \neq 0$.

I want to prove that for all $v \in V$ with $v \neq 0 \implies \exists f \in V^{*} : f(v) \neq 0$. I know that if $V$ is finite-dimensional we can choose a basis $\{e_i\}$ of $V$ and construct the corresponding dual basis $\{e^{*}_i\}$. If $v \neq…
jazzinsilhouette
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Definition of basis in infinite-dimensional vector space

I am struggling to understand the definition of a basis in an infinite dimensional vector space. Specifically, the definition I know says: A subset $B$ of a vector space $V$ is a basis for $V$ if every element of $V$ can be written in a unique way…
Powy
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the vector space of Magic Squares

Can anyone offer help? I have no clue how to do this problem. Magical squares are 3 by 3 matrices with the following properties: the sum of all numbers in each row, and in each column, and in each diagonal is equal. This number is called the magical…
stephenbutters
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Set of finite subsets as vector space: Double dual?

Some problem I've found while thinking about duals of vector spaces: Be $S$ an arbitrary set. Denote by $F(S)$ the set of finite subsets of $S$, and by $P(S)$ its power set. Now it is easy to see that $F(S)$ forms a vector space over…
celtschk
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Matrix of a conformal linear map

Could you please explain why every conformal linear map is a scalar times a rotation matrix? I can prove that every scalar-rotation matrix is a conformal map but not the opposite.
Amontillado
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Is a matrix with characteristic polynomial $t^2 +1$ invertible?

Given that $A$ is a square matrix with characteristic polynomial $t^2+1$, is $A$ invertible? I'm not sure, but this question seems to depend on whether $A$ is over $\mathbb{R}$ or over $\mathbb{C}$. My reasoning is that if $A$ is over $\mathbb{C}$…
Robert S. Barnes
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Is the product of square singular and non singular matrices always singular?

Given $A,B\in R^{n\times n}$ such that A is singular, and B is non-singular. Is $(AB)$ always singular? If so, how do I prove it?
Paul
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Is it possible to prove the Fundamental Theorem of Algebra for all polynomials of degree $n \le 4$?

Recenly I've been wondering whether it's possible to prove the FTA for all polynomials of degree $n \le 4$ without utilizing advanced maths but, at most, basic linear algebra (concepts such as eigenvectors, eigenvalues, determinants etc.). I've…
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