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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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Existence of a basis such that $\|e_i\|=1$ and $\|e_i^{*}\|_*=1$. (dual)

Let $E$ a $n$-finite dimensional normed vector space. Can we find a basis $e_1,e_2,\cdots,e_n$ of $E$ such that $\|e_i\|=1$ and $\|e_i^{*}\|_*=1$ for all $i$ ? where $\|\|_*$ is the dual norm. I know that $n=\dim(E)=\dim(E^*)$ and In the case of…
user117932
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Example of functions where linear dependence isn't obvious

The Wronskian lets us determine if a set of functions (possibly the solutions to a differential equation) are linearly dependent or not. But, for every example in the book, it is very obvious if one of the functions is a linear combination of the…
futurebird
  • 6,248
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Trace of the power matrix is null

Let $K$ be a field of characteristic $p \geq 0$ and let $M$ be a matrix $n \times n$ over $K$. If $p \nmid n$ and $Tr(M^i) = 0$ for all $i = 1,\dots,n$, how to prove that $M + Id_n$ is invertible? If $p=0$, we can write down a proof using the usual…
Hugo Rafael Oliveira Ribeiro
  • 353
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Wikipedia's proof of Schur Product Theorem

The Schur Product Theorem basically states that the Hadamard product of two semidefinite matrices is semidefinite. The proof from Wikipedia: ==== Proof of positivity ==== Let $M = \sum \mu_i m_i m_i^T$ and $N = \sum \nu_i n_i n_i^T$. Then $$ M…
user123276
  • 3,445
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Geometric Interpretation of Solutions to Linear Systems

I've been reading Linear Algebra by Jacob for self study and I'm wondering about the geometric interpretation of a unique solution to systems of equations in $\mathbb R^3$. For example, the homogeneous system $$ \begin{eqnarray*} x+5y-2z &=&0,…
CritChamp
  • 229
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Proving $V$ is isomorphic to $W$ iff $\dim V=\dim W$

Let $V$ and $W$ be two finite vector spaces over $F$. Prove that $V$ is isomorphic to $W$ iff $\dim V=\dim W$ I think I got the general approach but I don't think it's rigorous enough. $\Rightarrow$ Suppose $V$ is isomorphic to $W$ then there's a…
GinKin
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How to prove two equations in linear algebra

Given the following definition: How to proof these two equations? and PS: Actually, there are two proofs preceding the two(I have no problem with the following two), they are: Maybe they are hints on solving the latter two. I encounter this…
xiaohan2012
  • 391
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Find the projection of $b$ onto the column space of $A$

$A= \left[ {\begin{array}{ccccc} 1 & 1 \\ 1 & -1 \\ -2 & 4 \end{array} } \right] $ and $b = \left[ {\begin{array}{cccc} 1 \\ 2 \\ 7 \end{array} } \right] $ I use the formula $p = A(A^{T}A)^{-1}A^{T}b$ $A^{T}A = \left[…
Adrian
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Proving that $a$ is eigenvalue of $p(T)$ iff $a=p(\lambda)$ for eigenvalue $\lambda$ of $T$

The solution is below, I just do not understand why if: $p(T)-aI$ is not injective, then $T-\lambda_jI$ is not injective for some j either. Also, what does repeatedly applying T to both sides accomplish? Thanks!
user123276
  • 3,445
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A question about inner product and Gram-Schmidt process

Let there be the following bilinear form: $\int_0^1f(x)g(x)x\,dx$, which acts on the polynomials with degree $\leq2$. I needed to prove it's an inner product and then find an orthonormal basis. I needed to use Gram-Schmidt proccess. So, when I make…
Jim
  • 249
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Where is the contradiction?

My linear algebra book states the following: Let $$a_0 + a_1 z + \dots + a_m z^m = 0.$$ (Where $z$'s signify polynomials). If at least one of the coefficients was non-zero, then there would be at most $m$ distinct values of $z$ that would satisfy…
linalconfused
  • 53
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If a matrix multiplied by its transpose equals the original matrix, is it symmetric?

Here's the question: Prove: If ATA = A, then A is symmetric and A = A2 I tried to solve this by using inference. Assume A is symmetric, prove A = A2 If A is symmetric, then by definition ATA = A2. Since ATA = A, then A = A2. Assume A = A2,…
beatgammit
  • 153
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Let $U,V$ be subspaces of the vector space $W$. Show that if $U\nsubseteq V$ and $V\nsubseteq U$ then $U \cup V$ is not a subspace.

Let $U,V$ be subspaces of the vector space $W$. Show that if $U\nsubseteq V$ and $V\nsubseteq U$ then $U \cup V$ is not a subspace. I know that in order to be considered a subspace, the matrix addition and scalar multiplication operations must…
Alan
  • 91
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Proving that a set is/is not semilinear

A subset $X$ of $\mathbb{N}^n$ is linear if it is in the form: $u_0 + \langle u_1,...,u_m \rangle = \{ u_0 + t_1 u_1 + ... + t_m u_m \mid t_1,...,t_n \in \mathbb{N}\}$ for some $u_0,...,u_m \in \mathbb{N}^n$ $X$ is semilinear if it is the union of…
Vor
  • 690
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Finding an integer orthogonal basis

Say I have some non-orthogonal basis of some vector space that only have integer elements. Is it possible to find an orthogonal basis consisting of basis vectors with integer elements?
RobVerheyen
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