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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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Is a symmetric matrix characterized by the diagonal of its resolvent?

The resolvent of a square matrix $A$ is defined by $R(s) = (A-sI)^{-1}$ for $s \notin \operatorname{spect}(A)$. Is knowing the diagonal of $R(s)$ for all $s$ sufficient to recover $A$ when $A$ is symmetric? edit: a counter-example of two matrices…
roger
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Area preserving transformations

Suppose $A$ is a linear transformation from $R^3$ to $R^3$ and $|det(A)| = 1$. I know that $A$ is volume preserving, but is it also area preserving? For example, if $a$ and $b$ are two vectors in $R^3$ that span a parallelogram, is the area of this…
user5165
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Help Understanding Proof of Replacement Theorem?

Sorry if this is a trivial question. The book is Linear Algebra Done Right by Axler, page 25-26. Theorem: In a finite-dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every…
Mat.S
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Berkeley Problems in Mathematics 7.5.22

This is the problem: Let $A$ be a real symmetric $n \times n$ matrix with non negative entries. Prove that $A$ has an eigenvector with non-negative entries I looked at the answer key and don't quite understand it. In the expression containing max,…
Jan Lynn
  • 1,183
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Dimension of spaces of bi/linear maps

For $V$ a finite dimensional vector space over a field $\mathbb{K}$, I have encountered the claim that $$ \dim(\mathrm{Hom}(V,V)) = \dim(\mathrm{Hom}(V \times V, \mathbb{K})) $$ where $\mathrm{Hom}(V,V)$ denote the vector spaces, respectively, of…
AFX
  • 565
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Prove $\ker(T^n)=\ker(T^{n+1})$ and $\operatorname{range}(T^n)=\operatorname{range}(T^{n+1}).$

Let $V$ be an $n$-dimensional vector space over a field $F$ and $T$ an operator on $V.$ Prove $\ker(T^n)=\ker(T^{n+1})$ and $\operatorname{range}(T^n)=\operatorname{range}(T^{n+1}).$ Suppose $v \in \ker(T^n).$ Then $T^n(v) = 0,$ implying that…
St Vincent
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Can $ABA-BAB=I$?

Let $A,B\in M_{n\times n} (\mathbb{C})$. Is it possible that $ABA-BAB=I$? I came across this interesting problem as I was studying for an exam. I guess in the case when $A$ and $B$ commute we have $A(A-B)B=I$ and I am not sure if it can…
Galois
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Demonstration: If all vectors of $V$ are eigenvectors of $T$, then there is one $\lambda$ such that $T(v) = \lambda v$ for all $v \in V$.

Let $T: V \rightarrow V$ be a linear operator. I need to demonstrate that if all nonzero vectors of $V$ are eigenvectors of $T$, then there is one specific $\lambda \in K$ such that $T(v) = \lambda v$, for all $v \in V$. I understand that, if all…
João Daniel
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Linear independence under weird condition

This is a problem from a linear algebra textbook. Given a finite dimensional inner product space $V$ with orthonormal basis $e_1, \ldots, e_n$, show that if a list of vectors $v_1, \ldots, v_n$ satisfies $\|e_j - v_j\| < \frac{1}{\sqrt{n}}$ for all…
gus
  • 247
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Understanding of the proof of "Cayley-Hamilton thm"

I'm reading the proof of Cayley-Hamilton theorem and I got stuck on the proof. Let $T$ be a linear operator on a finite dimensional vector space $V$. If $f$ is the characteristic polynomial for $T$, then $f(T)=0$; in other words, the minimal…
cokecokecoke
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$AXB$ sort of decomposition?

Let $f: M_n(\mathbb{C}) \to M_n(\mathbb{C})$ be a $\mathbb{C}$-linear map (not necessarily an algebra homomorphism). Do there exist matrices $A_1, \dots, A_d \in M_n(\mathbb{C})$ and $B_1 \dots, B_d \in M_n(\mathbb{C})$ such that $$f(X) = \sum_{j =…
user223957
6
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2 answers

If $N$ is nilpotent then there exists $A$ such that $A^2=I+N$

Suppose $N\in M_{3\times 3}^{\mathbb{C}}$ is a nilpotent matrix. Prove that there exists $A\in M_{3\times 3}^{\mathbb{C}}$ such that $A^2=I+N$. Hint: find $A$ in the form $A=P(N)$ where $P$ is a polynomial in $\mathbb{C}[x]$. I really can't think of…
user265336
  • 317
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Diagonalize the matrix A (complex numbers)

Diagonalize the matrix A $A=\begin{pmatrix}1 & 2 & 4 \\3 & 5 & 2 \\2 & 6 & 1\end{pmatrix}$ So, i began the problem by finding the characteristic polynomial which was $λ^3-7λ^2-15λ-27$ using long division i got $(λ-9)(λ^2+2λ+3)$ so i used the…
Charlene
  • 687
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2 answers

Any set with more elements than the dimension of vector space is linearly dependent

Let $V$ be a vector space with dimension $n$ such that $\{v_1,\cdots,v_n\}$ is its basis. Take $A\equiv\{a_1,\cdots,a_p\}\subset V$ with $p>n$. How do can I show that $A$ is linearly dependent without having to do all those summations. Is there an…
Guilherme Salomé
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Prove that $\lambda$ is an eigenvalue of $A$ if and only if $\lambda$ is an eigenvalue of $A^T$.

Prove that $\lambda$ is an eigenvalue of $A$ if and only if $\lambda$ is an eigenvalue of $A^T$. I'm stucked here, i've approached the problem by looking at $\det(A-\lambda I)=0\iff\det(A^T-\lambda I)=0$. I tried some cases, and I can see it when…
user144809
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