# Tagged Questions

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 ...

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 ...
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 $\sigma: G \to \text{GL}(V \otimes R)$ ...
79 views

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 ...
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 ...
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..."...
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 ...
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 ...
39 views

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 ...
42 views

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 ...
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 ...
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 ...
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 ...
194 views

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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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? ...
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 $\alpha$-...
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 ...
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 ...
### 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 $\lim_{n}\|x_n-x\|=0\Rightarrow \lim_n\|T(x_n-x)\|=0$....