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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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Proof of an equivalence with respect to self-adjoint operators

I am trying to solve this problem but I fail in the converse part... Show that the product of two self-adjoint operators is self-adjoint if and only if the two operators commute. ($\Rightarrow$) If we suppose that $T,U,V$ are…
ViKaN-HND504
  • 227
4
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1 answer

Underdetermined homogeneous system of linear equations has always infinitely many solutions

I want to prove that an underdetermined homogeneous system of linear equations always has infinitely many solutions. I know that a homogenous system of linear equations always has the trivial solution (0,0,...,0). I also know that an underdetermined…
Rummelluff
  • 489
4
votes
3 answers

Prove\Refute: for every norm in $\mathbb{R}^n: \left \| x \right \|\leq \max (\left \| x+y \right \|,\left \| x-y \right \|)$

I need to prove or refute that for every norm in $\mathbb{R}^n$:$ \left \| x \right \|\leq \max (\left \| x+y \right \|,\left \| x-y \right \|)$. It's been quite a while since I studied Linear algebra 1. I tried to look for vectors $x$ and $y$ such…
Jozef
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4
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1 answer

Show that $L(V,W)$ is isomorphic to $\mathbb{R^{n\times m}}$

Suppose $V$ and $W$ are vector spaces with $\dim V = m$ and $\dim W = n$. Show that $L(V,W)$ (the set of all linear transformations from $V$ to $W$) is isomorphic to $\mathbb{R^{n\times m}}$. How do you typically prove that something is…
Hailey
  • 141
4
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1 answer

Determine whether the set of vectors is linearly dependent or not

Suppose I have the vectors $\underline{a}_1, \underline{a}_2,\ldots,\underline{a}_k$ and $\underline{b} \neq 0$ in $\mathbf{R}^n$. Also, $\underline{a}_1 \neq \underline{a}_2 \neq \ldots\neq \underline{a}_k$. Say that the equation $x_1…
user187240
  • 45
4
votes
1 answer

Given the frame $B=\{(1,1,0),(0,1,1),(1,1,1),(0,0,1),(0,1,-1)\}$, find (if possible) a (i) $(2,1)$ surgery,

Given the frame $B=\{(1,1,0),(0,1,1),(1,1,1),(0,0,1),(0,1,-1)\}$, find (if possible) a (i) $(2,1)$ surgery, and a $(1,2)$ surgery that produce tight frames. A frame is tight if and and only if the frame operator $B^TB$ is a multiple of identity. So,…
Username Unknown
  • 4,633
4
votes
2 answers

Show that a matrix that maps orthonormal vectors to orthonormal is orthogonal

If $Av_1$ $Av_2$ ... $Av_k$ are orthonormal vectors in $R^n$ and $v_1$ $v_2$ ... $v_k$ are also orthonormal vectors in $R^n$. Show that the Matrix A must be orthogonal i.e. $A^TA=I$. I can prove it the other way round but can't this way. Thanks for…
shaktiman
  • 107
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1 answer

Why such a vectors are linearly independent?

Assume that $A$ is an linear operator on a real vector spacev $V$. I wish to prove that if for some $x,y \in V$ such that $x\neq 0$ or $y \neq 0$ and some $a,b \in \mathbb R$ and $b\neq 0$, the following conditions hold $$ Ax=ax-by,…
user172903
  • 205
4
votes
1 answer

Finite order endomorphisms

Why the finite order endomorphisms are diagonalizable over the complex numbers (or any algebraically closed field where the characteristic of the field does not divide the order of the endomorphism) with roots of unity on the diagonal?
Matt
  • 43
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7 answers

Proof that $-v = (-1)*v$

I need to prove that for every Vector Space this is valid: $$ -v = (-1)*v $$ -v = inverse element of addition -1 a real number $*$ the multiplication by real number of the Vector Space My teacher said that $-v$ is just a notation for the inverse…
Miranda
  • 41
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votes
1 answer

Multiplicity as roots of the minimal polynomial

Let $V\neq\{0\}$ be a finite-dimensional vector space over a field $F$ and let $\alpha \in \text{End}(V)$. Suppose that $\lambda$ is an eigenvalue of $\alpha$ with multiplicity $r$ as a root of the minimal polynomial of $\alpha$. I want to show that…
Elliptic2005
  • 789
4
votes
2 answers

Bounds on the singular values of a matrix with unitary columns

if $X$ is a matrix with unitary columns ( each column has unit norm ), are there lower and upper bounds on the minimum and maximum singular values of $X$? I could prove a lower bound for $\Sigma_{min}$ and $\Sigma_{max}$ i.e. $\Sigma_{min} \geq 0$…
Karthik Upadhya
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  • 15
4
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1 answer

Image of the matrix and its rref

Consider a matrix $A$. Is $ \mathrm{image}(A) $ equal to $ \mathrm{image}(\mathrm{rref}(A)) $? Their kernels and solution of the system $ Ax = b $ is the same.
Mohamed Osama
  • 263
4
votes
2 answers

Finding a particular solution to a non-homogeneous system of equations

If one asked to solve the set of equation below with the associated homogenous system, I'd know how to do it. $$S \leftrightarrow \begin{cases} 3x + 5y + z = 8\\\ x + 2y - 2z = 3 \end{cases}$$ $$S' \leftrightarrow \begin{cases} 3x + 4y + z =…
user21385
4
votes
2 answers

Sum and intersections of vector subspaces $U_1+U_2=(U_1 \cap U_2) \oplus W$

Let $U_1,U_2$ be vector subspaces from $\in \mathbb R^5$. $$\begin{align*}U_1 &= [(1,0,1,-2,0),(1,-2,0,0,-2),(0,2,1,2,2)]\\ U_2&=[(0,1,1,1,0),(1,2,1,2,1),(1,0,1,-1,0)] \end{align*}$$ (where [] = [linear span]) Calculate a basis from $U_1+U_2$ and…
Clash
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