Mathematics Stack Exchange
Mathematics Stack Exchange

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
6
votes
1 answer

Geometric and analytic multiplicity of a linear operator

If I understand correctly, the analytic multiplicity of a linear operator say $T:V\to V$ is the amount of times $\lambda$ shows up as a root in the characteristic polynomial (Assuming you have a matrix $A$ representing $T$ with respect to some basis…
Justin
  • 2,813
6
votes
4 answers

Linear Transformation

Alright so I have this Transformation that I know isn't one to one transformation, but I'm not sure why. A Transformation is defined as $f(x,y)=(x+y, 2x+2y)$. Now my knowledge is that you need to fulfill the 2 conditions: Additivity and the scalar…
user432184
  • 61
6
votes
2 answers

are there any matrices that act like the identity matrix but are totally different?

Given any matrix $A$ or vector $v$, can you find a matrix $B \neq I$ and also not resembling $I$ (having entries in places other than diagonal) such that $B*A$ and $B*v$ are approximately $A$ and approximately $v$? For me the answer is should…
Nażysław Zbyłutowicz
  • 709
6
votes
2 answers

Linear Algebra: Projection Maps

I would like to check if my understanding of projection maps is correct. I have been given the following subset of $\mathbb{R}^3$: $$A=\left\{\begin{pmatrix} x \\ y \\ -x+2y \end{pmatrix} \middle| x,y,z\in\mathbb{R}\right\}$$ A basis for this subset…
Jon
  • 61
  • 1
  • 3
6
votes
2 answers

Definition of the characteristic polynomial

My textbook defines the characteristic polynomial of an $n \times n$-matrix $A$ as: $$p_A = \text{det}(A-XI_n) = \sum_{\sigma \in S_n} \text{sign } \sigma. (a_{1\sigma(1)}-X\delta_{1\sigma(1)})...(a_{n\sigma(n)}-X\delta_{n\sigma(n)})$$ (where $S_n$…
Haidar Abboud
  • 179
6
votes
4 answers

To show $\mathrm{ker} f=\{0\}$ for linear mapping $f$.

Let $V$ be a vector space over $F$ with basis $\{e_1,e_2,...e_n\}$. Let $F$ be a linear mapping from $V$ to $V$ such that $F(e_1) =e_2,...F(e_n)=e_1$. Show that $\mathrm{ker} f=\{0\}$. Also find $f^{-1}$. I just know that $\mathrm{ker} f=\{0\}$ iff…
Kavita
  • 728
6
votes
4 answers

Which symbol can be used to refer to identity matrix?

$I$ is commenly used as a notation of identity matrix. I am wondering is there any notation else for identity matrix?
aban
  • 215
6
votes
1 answer

Matrices commuting with diagonal matrix, 3 distinct diagonal entries.

Hopefully I have the right idea? Let $A\in\mathbb{C}^{4\times 4}$ be a diagonal matrix with exactly 3 distinct entries on its main diagonal. What is the dimension of the vector space over $\mathbb{C}$ of matrices $B\in\mathbb{C}^{4\times4}$ such…
user346096
  • 1,068
6
votes
2 answers

Every cyclic subspace contains an eigenvector

The question is : Let $X$ be a non-null vector.Then there exists an eigenvector $Y$ of $A$ belonging to the span of $\{X, AX, A^{2} X, ... \}$. I have tried to the best of my ability to solve it. But I don't find any right way to proceed.Please…
Arnab Chattopadhyay.
  • 2,570
6
votes
1 answer

Finding vector subspaces

I've got this problem: Let $H = \left \{ x \in \mathbb{R}^{4} \, \left| \, x_2 - x_3 + 2x_4 = 0 \right. \right \}$ Find, if possible, $a \in \mathbb{R}$ and $S, T$ vector subspaces so that $\dim(S) = \dim(T)$, $S + T^\perp = H$, $S \cap T = \left…
Lucas
  • 133
6
votes
1 answer

The largest element in magnitude of a Hermitian positive definite matrix is on the diagonal

Anyone can help provide a proof for The largest element in magnitude of a Hermitian positive definite matrix is on the diagonal. I found one related question Why is the largest element of symmetric, positive semidefinite matrix on the diagonal?…
River
  • 925
6
votes
1 answer

Distributivity of subspaces

I need to either give a proof or find a counterexample to a statement: $$L+(M∩N) = (L + M)∩(L + N)$$ Where $L$,$M$,$N$ are subspaces of a vector space $V$. I could do $LHS⊆RHS$ proof, but I'm stuck with backwards proof. I would be really grateful…
June15
  • 107
6
votes
1 answer

common eigenvectors of commuting operators

I am pretty sure that my problem has already discussed, but I didn't find. So, the question is how to prove that two commuting operators have a common eigenvector. The first note is following: Let $A$ be the operator such that $Av = \lambda v$ (we…
Invincible
  • 2,572
  • 10
  • 26
6
votes
1 answer

Dimension of an object

I know the dimension of a straight line, a circle is 1. But how can we prove it by using vector space? The dimension a vector space is the number of elements in the basis.
Prince Khan
  • 1,544
6
votes
3 answers

Prove: The set of all polynomials p with p(2) = p(3) is a vector space

Prove that this set is a vector space (by proving that it is a subspace of a known vector space). The set of all polynomials p with p(2) = p(3). I understand I need to satisfy, vector addition, scalar multiplication and show that it is non…
student101
  • 567
1 2 3
…
99 100