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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 $A^3B = BA^3$, then $AB = BA$.

Let $A$ be a Hermitian matrix. Suppose there exists a matrix $B$ such that $A^3B = BA^3$. Show that $AB = BA$. I was trying to use the fact that since $A$ is Hermitian, there exists a unitary matrix $U$ such that $UDU^* = A$, thus $UD^3U^* B =…
algor207
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6
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3 answers

If $X + X^T$ is positive definite, is $X^{-1} + X^{-T}$ also positive definite?

Is it true or is there a counterexample?
Yuan Gao
  • 597
6
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1 answer

linear functionals linearly independent

Let $V$ be a vector space with $\dim V=n$. Let $\varphi_1,...,\varphi_n $ be linear functionals that are not $0$. Prove that $\varphi_1,...,\varphi_n $ are linearly independent if and only if $\cap_{i=1}^n \ker \varphi_i = \{0\}$. $\\$ I succeeded…
Shlomi
  • 811
6
votes
3 answers

Trace of a matrix $A$ with $A^2=I$

Let $A$ be a complex-value square matrix with $A^2=I$ identity. Then is the trace of $A$ a real value?
Tama
  • 71
6
votes
2 answers

Prove that: if $T$ is an irreducible linear operator then $T$ is cyclic

Let $T:V\to V$ be a linear operator on a finite dimensional vector space $V$. I need to prove that: If $T$ is irreducible then $T$ is cyclic My definitions are: $T$ is an irreducible linear operator iff $V$ and {$0$} are the only complementary…
user128422
  • 3,057
6
votes
4 answers

How to prove that the dimension of a hyperplane is n-1

The hyperplane $H$ defined by $$H:=\{x\in\mathbb {R}^n:a^Tx=b\}$$ is the set that has dimension $n-1$, my question is why or how can we prove that its dimension is $n-1$? Thank you to every one who provide any help or if possible the proof for that.
rinan yosf
  • 129
6
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2 answers

$3$-linear functions on $\mathbb{R}^3$

I know that any bilinear function $\Phi$ can be presented in a unique way as a sum $$\Phi = S + A,$$ where $S$ is a symmetric and $A$ is skew-symmetric bilinear functions. Does a similar statement hold for $3$-linear functions on $\mathbb{R}^3$?
user198822
6
votes
9 answers

Inverse of a Product of Matrices: Why is $(AB)^{-1} = B^{-1} \cdot A^{-1}$?

I'm having a bit of difficulty conceptualizing a rule for the inverse of a product of matrices and I'd appreciate any input on it. Suppose I let: $A^{-1} = \begin{bmatrix}1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B^{-1} = \begin{bmatrix}5 & 6 \\ 7 & 8…
user153085
6
votes
4 answers

The definition of $\oplus$

I would like to understand why the books give two different concepts to $\oplus$ between vector spaces: See: Concept 1: $W=V_1\oplus V_2=\{(v_1,v_2) \mid v_1\in V_1, v_2\in V_2\}$. Concept 2: $W=V_1\oplus V_2=\{v_1+v_2 \mid v_1\in V_1, v_2\in…
user42912
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6
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1 answer

Prove that the largest singular value of a matrix is greater than the largest eigenvalue

Let $\sigma_1$ be the largest singular value of the matrix $A = (a_{ij})$. Show that $\sigma_1 >= \lambda_{max}$, where $\lambda_{max}$ denotes the largest eigenvalue of $A$, and that $\sigma_1 \geq |a_{ij}|_{max}$. $A$ must be a square matrix,…
Blackeyes
  • 461
6
votes
3 answers

number of elements on a finite vector space

I know that there is no vector space over any field $\mathbb{F}$ having precisely $6$ elements. I would like to know if there is a theorem which characterizes those $n's \in \mathbb{N}$ such that there is a vector space having exactly $n$ elements…
user23505
6
votes
1 answer

Sparseness of a Vector

The sparseness of a vector is defined a follows: $$\psi(\textbf{x}) = \frac{\sqrt{n}-\frac{\left(\sum_{i} x_i \right)}{\sqrt{\sum_{i} x_{i}^{2}}}}{\sqrt{n}-1}$$ So $\psi(\textbf{x}) =0 $ if $\sqrt{n} = \frac{\left(\sum_{i}x_i \right)}{\sqrt{\sum_{i}…
Robbie
  • 61
6
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1 answer

Why aren't all real self-adjoint operators diagonal?

I'm experiencing some confusion regarding self-adjoint operators. As background for my question, I give the following 3 results (all from Linear Algebra, 3rd ed. by Friedberg, Insel, and Spence): Theorem 6.17: "Let $T$ be a linear operator on a…
Alex Petzke
  • 8,763
5
votes
1 answer

Prove that the linear transformation is injective iff $T(f_1),\ldots,T(f_n)$ are linearly independent

$V$ and $W$ are vector spaces with $dim(V)=n$. Prove that a linear transformation $T:V\rightarrow W$ is injective if and only if for a basis $B=(f_1,\ldots,f_n)$ of $V$, $T(f_1),\ldots,T(f_n)$ are linearly independent. I'm first trying to…
Hailey
  • 141
5
votes
1 answer

Why is not a vector space isomorphic to its dual space?

Let $V$ be a finitely generated vector space with a basis $\mathcal{B}=\{\alpha_1,\cdots,\alpha_n\}$ and let $\mathcal{B}^*= \{f_1,\cdots,f_n\}$ be the dual basis of $\mathcal{B}$. In this situation, I defined a function $T:V\to V^*$ with…
Analysis
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