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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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Can an upper triangular matrix be put in jordan form using upper triangular matrices and permutations only.

Give an upper triangular matrix A does there always exist $P, P^{-1}$ and $U, U^{-1}$ such that $PUAU^{-1}P^{-1}$ is in Jordan canonical form and U is upper triangular and P is a permutation matrix. I believe the answer is yes as when we have…
Faust
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Proof of "every finite dimensional vector space has a finite basis"

Finite dimensional implies the existence of a finite set that spans the vector space. Let V be one such vector space and let S be a finite set that spans V. The text I am following has a theorem that Theorem 1: Any minimal spanning set of V is a…
TryingHardToBecomeAGoodPrSlvr
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Given a matrix $A$ find a matrix $C$ such that $C^3$=$A$

This is a question I had on a test, we were told not to use brute-force and figure out a smart way to solve the problem. We have a matrix $A =$ $\displaystyle\begin{bmatrix} 2 & 3\\ 3 & 2 \end{bmatrix}$. Find a matrix $C$ such that…
user8408
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The `square root of a tensor'; conditions of existence.

Suppose I have some tensor, for concreteness I will consider a rank 4 tensor with components $R_{abcd}$ where the indices run over 0,1,2,3 (my question arises in a physics context). Under what conditions is the existence of another tensor, call it…
NormalsNotFar
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Tensor product of two vector spaces

There is something which always intrigue me. Let $U$ and $V$ be vector spaces over $k$ a field. Is it true that if $U\otimes_{k}V=0$, then $U=0$ or $V=0$. Note that $U$ and $V$ are not necessarily finite dimensional. I know that if $\{u\}$ and…
enoughsaid05
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linear independent over $\mathbb{Q}$

Let $r_1, r_2, \cdots, r_n$ be distinct rational numbers in the interval $(0,1)$. How to prove that in the space $\mathbb{R}$ over $\mathbb{Q}$ the numbers $2^{r_1}, \cdots, 2^{r_n}$ are independent?
SWalker
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What is the rank of the cofactor matrix of a given matrix?

Let $A = (a_{ij}) ∈M_n(\mathbb{R})$; $n≥3$. Let $B = (b_{ij})$ be the matrix of its co- factors, i.e. $b_{ij}$ is the cofactor of the entry $a_{ij}$ in $A$. What is the rank of $B$ when a. the rank of $A$ is $n$? b. the rank of $A$ is less than, or…
ketu
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What is rank of $f(A)$, where $f$ is the minimal polynomial of $A$?

If $f(x)$ is minimal polynomial of the $4\times 4$ matrix $$A=\begin{pmatrix} 0 &0 &0& 1\\ 1 &0 &0 &0\\ 0 &1 &0 &0\\ 0 &0 &1 &0 \end{pmatrix}$$ Then what is rank of $f(A)$? I think $f(A)$ will be a zero matrix so its rank is 0. Am I right?
Alka Goyal
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Nullity and rank bounds for a nilpotent matrix

Let $A=\mathbb R^{11}\to \mathbb R^{11}$ be a linear transformation such that $A^5=0$ and $A^4\neq 0$. Which of the following is true? a) $\operatorname{null}A\le7$ b) $2\le\operatorname{null}A$ c) $2\le\operatorname{rk}A\le9$ I don't know…
RAM_3R
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How to show that the following eigenvectors have to be orthogonal?

I have the following problem: Suppose that $A$ is a symmetric matrix, with $A$ = $A^{T}$ . Suppose $\vec{v}$ and $\vec{w}$ are eigenvectors of $A$ associated with distinct eigenvalues. Show that $\vec{v}$ and $\vec{w}$ must be orthogonal. (Hint:…
user7997
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Why does multiplying by $\textbf{A}^T$ make a previously unsolvable linear system solvable

Consider for instance the linear system: $$\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6 \\ \end{array} \right).\left( \begin{array}{c} x \\ y \\ \end{array} \right)=\left( \begin{array}{c} 1 \\ 2 \\ 4 \\ \end{array} \right)$$ This is…
1110101001
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A be a $3\times3$ real valued matrix such that $A^{3}=I$ but $A \neq I$ .Then trace(A)=?

I was thinking about the following problem: Let A be a $3\times3$ real valued matrix such that $A^{3}=I$ but $A \neq I$ . Then trace of A must be (a)0, (b)1, (c)-1, (d)3. My attempts: I take A to be $$\begin{pmatrix} x &0 &0 \\ 0& x &0…
learner
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Prove that idempotent operator $E$ is self-adjoint if and only if $EE^∗$ = $E^∗E$

Let $V$ be a finite-dimensional inner product space, and let $E$ be an idempotent linear operator on $V$, i.e., $E^2 = E$. Prove that E is self-adjoint if and only if $EE^* = E^*E$. Are there any simpler answers to the question that the answers…
A. Napster
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How to normalize the matrix?

I have this matrix, \begin{equation} T=\begin{bmatrix}a&b\\-b&-a\end{bmatrix} \end{equation} To normalize it, the matrix $T$ must satisfy this condition: $T^2=1$ and $1$ is the identity matrix. To solve that I set $x^2T^2=1$ and solve for x which is…
Aschoolar
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Let $T:V→V$ be a linear transformation satisfying $T^2(v)=-v$ for all $v\in V$. How can we show that $n$ is even?

Let $V$ be a real n dimensional vector space & let $T:V→V$ be a linear transformation satisfying $T^2(v)=-v$ for all $v\in V$. Then how can we show that $n$ is even? I am completely stuck on it. Can anybody help me please?
abdakchi
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