For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag:linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces ...

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13
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431 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
5
votes
0answers
71 views

Construction of vector space isomorphism where $f(v \otimes gh) = \sigma(g)(f(v \otimes h)),\text{ }\forall v \in V,\text{ }g,\,h \in G$

Let $G$ be a finite group, $R = \mathbb{C}G$ the regular representation of $G$, and $\rho : G \to \text{GL}(V)$ a finite dimensional representation of $G$. Write $\sigma: G \to \text{GL}(V \otimes R)$ ...
5
votes
0answers
79 views

Square root of differentiation

Let $T=d/dx$ be the differentiation operator on vector space $V=C^{\infty}(\mathbb{R})$, the space of real (complex) valued smooth maps on real line. To what extent, all subvector space $W\...
5
votes
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149 views

Eigenvalues of a difference of non commuting products

Let $A$ be an $n \times n$ complex matrix and let $T(M) = AM-MA$. I need to determine the eigenvalues of $T$ in terms of those of $A$. This was an exercise from Artin, and I was not being able to ...
5
votes
0answers
180 views

Is the Laplace transform a vector space isomorphism? And what space is it isomorphic to?

The laplace transform is a linear transformation, $\mathcal{L}: \mathcal{M} \rightarrow?$, where $\mathcal{M}$ is the set of exponentially bounded functions on $\mathbb{R},$since $\mathcal{L}(af(x)+bg(...
5
votes
0answers
87 views

Mapping vector spaces over two different fields?

I was having linear algebra class and we have been discussing about a possible group homomorphism that might allow mapping between two vector spaces over two different fields This is also an ...
5
votes
0answers
91 views

Decomposition of order-$n$ tensors

If $V$ is a finite-dimensional vector space, then $V\otimes V\cong\mathbf{S}^2(V)\oplus\bigwedge^2(V)$. The first summand on the right is the symmetric part of $V\otimes V$ and the second summand is ...
5
votes
0answers
162 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like "intuitively..."...
5
votes
0answers
123 views

Generating a 3d ribbon from a series of points

I am attempting to generate a 3d ribbon from a set of 3d points. The idea is to generate a realistic ribbon which follows those points. In its current state, one example looks like this: In this ...
5
votes
0answers
232 views

Are there eigenvectors, eigenvalues, and characteristic functions for non-linear vector fields?

An eigenvector is a vector in the preimage of the transformation whose direction is not changed when the transformation is applied. It seems like the concept of eigenvectors and eigenvalues would ...
4
votes
0answers
39 views

scalar product for $\mathbb Q$-vector space in $\mathbb C$

In the texbooks I have for linear algebra, the scalar products are only introduce for $\mathbb R$ and $\mathbb C$ vector space, that lead me to following question: $W:=span(1,\sqrt{2}) \subset \...
4
votes
0answers
47 views

On kernels of commuting operators in infinite dimensions

Let $X$ be an infinite dimensional vector space, and let $\operatorname{S},\operatorname{T}\in\mbox{End}(X)$ be two operators such that: $\operatorname{T}\operatorname{S}=\operatorname{S}\...
4
votes
0answers
46 views

Name for “Almost a vector space, but with $\mathbb{N}_0$ instead of a field”

I have a finite set of vectors $V\subset \mathbb{R}^n$ Let us enumerate $V = \{\tilde{v}_1, \tilde{v}_2,...,\tilde{v}_m\}$ I have some space that I want to talk about (I spend a lot of time talking ...
4
votes
0answers
42 views

When is $P_\mathcal S ACA'Q_\mathcal S = 0$?

All matrices in this question are real. Let C be a positive definite matrix and $A$ an arbitrary square matrix such that the product $ACA'$ makes sense. Let $\mathcal S$ be a space such that $R(A')\...
4
votes
0answers
133 views

The topology of $GL(V)$

Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
4
votes
0answers
192 views

The dimension of vector space $F^{X}$.

Here's the problem: Let $F$ be field, $X$ an infinite set and $F^{X}$ be the set of all functions $f:X\rightarrow F$. Then $F^{X}$ is a vector space over $F$ (with $(f+g)(x)=f(x)+g(x)$ and $(rf)(x)=...
4
votes
0answers
220 views

How to visualize cotangent spaces.

I was wondering how to intuitively and visually understand dual vector spaces and one-forms. So my question is (1), how to visualize cotangent spaces and (2), how to intuitively understand them? My ...
4
votes
0answers
59 views

Blocking set for cosets of codimension $2$

In this paper following theorem is proved: If $V$ is vector space of dimension $n$ over a finite field $F$ of $q$ elements then any subset of $V$ which meets every hyperplane of $V$ contains at least ...
4
votes
0answers
97 views

Structure of a fuzzy subspace

Let $V$ be a vector space over a field $F$ and let $f$ be a function from $V$ to the interval $I:=[0,1]$ satisfying the condition that for any $a \in I$ the set $V_a:=\{v \in V | f(v) \ge a\}$ is a ...
4
votes
0answers
600 views

Show that if $S$ is a subspace of a vector space $V$, then $\dim (S) \leq \dim (V)$

Show that if $S$ is a subspace of a vector space $V$, then $\dim(S) \leq \dim(V)$. Furthermore, if $\dim(S)= \dim(V) < \infty$ thn $S=V$. Give an example to show that the finiteness is required in ...
4
votes
0answers
194 views

Vandermonde matrix

Let ${\bf G} \in\mathbb{C}^{M\times K}$ and ${\bf H} \in\mathbb{C}^{N\times K}$ are full-rank Vandermode matrices where $MN-1=K>N\geq N$, that is, ${\bf G}$ and ${\bf H}$ are fat. Let ${\bf F}= {\...
3
votes
0answers
34 views

Series and Linear Algebra

I have been studying series and I noticed that the Taylor series converges to a function using a polynomial basis and the Fourier series converges to a function in a given interval using a ...
3
votes
0answers
30 views

Sum of projection operators

Given $p_1, ..., p_n$ $n$ projection operators on the vector space $E$ such that $\sum_{i=1}^n p_i$ is a projection operator. How to show that $\forall i,j \text{ s.t. } i \neq j, p_i \circ p_j = 0$ ?...
3
votes
0answers
38 views

Color space as a vector space

I am not sure that this is the best place for this topic, so I apologize in advance. I have two questions. I think that color space with say additive colors (red, green, blue) forms a vector space. ...
3
votes
0answers
43 views

Proving a trivial bound on $L_2$ norm of the error in a sparse approximation of a vector

Trying to understand this supposedly 'trivial' bound from a paper: If $\theta_N$ denotes the vector $\theta$ with everything except $N$ largest coefficients set to $0$ then we have $$ || \theta - \...
3
votes
0answers
136 views

Volume of $n$-dimensional spherical orthant in upper diagonal halfspace

Consider an $n$-dimensional Euclidean Space. Consider orthants in that space. Each orthant occupies $\frac{1}{2^n}$ of the volume of an $n$-dimensional unit sphere. Let's call that a spherical ...
3
votes
0answers
41 views

Direct sum in vector spaces with arbitrary dimension

Let be $V$ a vector space, with arbitrary dimension, and $Z\subset V$ proper subspace. Is it always possible write $V$ as a direct sum, $$V=Z+W?$$ Proof: Let be $\{z_\alpha\}_{\alpha\in \lambda}$ a ...
3
votes
0answers
54 views

Problem with the proof of the undetermined coefficients method

I'm trying to proof the following property which is the base of undetermined coefficients method. Say I have a differential equation of the form: $$x^{(n)}+a_{n-1}x^{(n-1)}+\cdots+a_0 = f(t)$$ When ...
3
votes
0answers
34 views

There exists an algebraic basis $(e_i)_{i \in I}$ in $E$ such that $\|e_i\| = 1$ for all $i \in I$?

Let $E$ be a infinite-dimensional normed vector space. How do I see that there exists an algebraic basis $(e_i)_{i \in I}$ in $E$ such that $\|e_i\| = 1$ for all $i \in I$?
3
votes
0answers
21 views

generalizations of the linearly independence of column vectors in a Vandemonde matrix to higher dimensions

Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $ or $\mathbb{C}$. By the non-degeneracy of Vandemonde matrix the maps $$ f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$ x\longmapsto (1,x,x^2,\cdots,x^{n-1}...
3
votes
0answers
49 views

Could we talk about affine spaces before vector spaces?

I was recommended the book Geometry by Michele Audin by a professor when I asked about learning more about affine geometry. I like the book, but it's raised a question. To me, it seems that it would ...
3
votes
0answers
28 views

Formulas give irreducible representation, $SL(2, \mathbb{C})$.

Let $\text{U}$ be an associative $\mathbb{C}$-algebra with three generators $E$, $H$, $F$, and three defining relations$$HE - EH = 2E,\text{ }Hf - FH = -2F,\text{ }EF - FE = H.$$Let $\mathbb{C}^m[x, y]...
3
votes
0answers
72 views

Constrained optimization over vectors

Edit: Even strategies for attack (as opposed to complete answers) are appreciated-- this is a hard question. I have a function $f(v,p)$ where v is a vector of length $n$, and I'm trying to derive ...
3
votes
0answers
46 views

basis for $\mathbb{R}^{\mathbb{N}}:=\left\{f:\mathbb{N}\to\mathbb{R}\right\}$, and its cardinality.

I know that all vector space has a basis. My question is "concrete" example for basis for $\mathbb{R}$-vector space $\mathbb{R}^{\mathbb{N}}:=\left\{f:\mathbb{N}\to\mathbb{R}\right\}$. If I refer ...
3
votes
0answers
65 views

Working in a field.

When I calculate vector spaces, diagonalization of matrices, linear transformations on a field, can I work in $\mathbb R$ and ultimately transform the result to that field? For example if I calculate ...
3
votes
0answers
25 views

$N$-dimensional linear operator is normal, Lagrange interpolation?

Is there a way to see that an $N$-dimensional linear operator $A$ is normal if and only if $A^\dagger$ can be represented as a linear combination of $I, A, A^2, \dots, A^{N-1}$ using Lagrange ...
3
votes
0answers
40 views

Question regarding example of toric variety and generators of cone

Consider the canonical example taking n=2, and taking the cone $\sigma$ generated by the vectors $e_{2}$ and $2e_{1} - e_{2}$. The dual cone $\sigma^{v}$ is defined as the set of vectors in the dual ...
3
votes
0answers
61 views

Basis of $\mathbb{F}[[x]]$ over $\mathbb{F}$ without AC

Does the ring of formal power series $\mathbb{F}[[x]]$ as a vector space over $\mathbb{F}$ admit a basis without assuming the Axiom of choice, at least in some special cases of $\mathbb{F}$? I'm ...
3
votes
0answers
28 views

Is this the correct solution involving vector subspaces and basis?

I need to find the basis and hence dimension of a subspace of $\mathbb{R^3}$. 1) $$U=\{(x,y,z):x=2y\}$$ Solution: We have $x=2y \iff y=\frac{x}{2}$ therefore we can write all elements in $U$ as the ...
3
votes
0answers
125 views

Matrix which is not similar to it's transposed

Let $V$ be vector space over a field $\mathbb{k}$. I can prove that any matrix is similar to its matrix transpose if $\mathbb{k}$ is an infinite field, but is this still true when $\Bbb k$ is finite? ...
3
votes
0answers
54 views

Vector spaces and infinite cyclic linear transformations

Let $G$ be a direct sum of infinitey many copies of the group of rational numbers $\mathbb{Q}$. Let $\alpha$ be an automorphism of $G$ with infinite order. Is always possible to find an $\alpha$-...
3
votes
0answers
29 views

Free ordered vector space over an ordered abelian group

Let $G$ be a partially ordered abelian group (written additively). I want to add $\mathbb{R}$-multiples to $G$ in a "free" way ,thus extending $G$ to an ordered vector space. Construction: To this ...
3
votes
0answers
52 views

Add vectors from a set to reach the goal vector, using the minimum possible cost

I am trying to solve a problem in an optimal way. The problem is as follows: We have an n-dimensional space In this space, we have a "finish" point with n coordinates, all non-negative We have a set ...
3
votes
0answers
131 views

$\rm span(S_1) + \rm span(S_2) = \rm span(S_1 \cup S_2)$ for infinite sets

I have these two definitions of span: Span: Suppose a vector space $(V,+,\cdot)$, and $$S = \{u_1,\cdots,u_n\}$$ (and $S$ is a subset of $V$, not a subspace) $$[S]=:\cap_{w\...
3
votes
0answers
71 views

Find the projection of any vector onto the linear span and the normal from any vector to that span

Show that the vectors $u_1 = (1/9,4/9,8/9), u_2=(8/9,-4/9,1/9), u_3=(-4/9,-7/9,4/9)$ form an orthonormal basis of $\mathbb{R}^3$. Find the projection of any vector $x=(\xi_1,\xi_2,\xi_3) \in \mathbb{...
3
votes
0answers
50 views

Notation for a vector space: $(\mathbb{C}^\infty)^{\otimes L}$

In a paper, the authors use the notation $(\mathbb{C}^\infty)^{\otimes L}$, where $L$ is a constant, for a vector space, but they do not give a definition. They also implicitly introduce an inner ...
3
votes
0answers
60 views

Higher Dimensional Right-Hand Rule

In seven dimensions, the cross product makes sense. Without resorting to nonvector tensors or exterior products (although they can be used to further explain), how does one perform this cross product ...
3
votes
0answers
64 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
3
votes
0answers
78 views

Find closest vector to a given vector from a particular set of vector

Let $x=\left(x_t\right)_{t=1}^n$ be a vector such that $$ x_t = \prod_{i=1}^t u_i, \tag{1} $$ where each parameters $u_i$ can take any of two value $$ u_i \in \left\{a,b \right\} = \left\{ 1.3, 0.8 \...
3
votes
0answers
153 views

Linear Operators: Continuous $\Rightarrow$ Bounded

Let $T:V\rightarrow V'$ be a continuous linear operator between two normed vector spaces $V,V'$. Show that it is bounded. Continuity is defined as $\lim_{n}\|x_n-x\|=0\Rightarrow \lim_n\|T(x_n-x)\|=0$....