# Tagged Questions

84 views

### An Algebraic Version of vector spaces

Consider the following set of real numbers $\mathcal{X}=\{1,2,3,\sqrt{2}+1,\pi+\sqrt{2}\}$. Lets consider the set of all linear combinations with integer coefficients of these numbers which I will ...
33 views

### What is the difference between a module of finite rank and finitely generated module.

R is an integral domain and every module we talk about is an R-module. If a module is finitely generated then obviously every element of the module can be written as finite R-linear combination of the ...
39 views

### Find a basis of the $k$ vector space $k(x)$

Suppose $x$ is a transcendental over field $k$ and $k(x)$ is the field of fractions of $k[x]$. Can we explicitly express a basis of the $k$ vector space $k(x)$？
36 views

### $\operatorname{rank}(F) = \operatorname{dim}_{k}(\frac{F}{mF})$

Let $R$ be a commutative ring with unit; $m$ is a maximal ideal; $F$ a free $R$-module. We know that $\frac{F}{mF}$ is a vector space over $\frac{R}{m} = k$ . I have to prove that ...
59 views

### Proof about finite dimensional vector spaces over fields

Prove that every finite dimensional vector space $V$of dimension $n$ over a field $F$ is isomorphic to the vector space $F^n$. Okay, lot's of stuff here. I think most of the reason I can not do this ...
85 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 ...
47 views

### Finding the dimension of a vector subspace

Consider $\mathbb{F}_{2}^{n} = \{(k_{1}, k_{2}, ... , k_{n}) : k_{i} \in \{0,1\}$ mod $2\}$. Let $M$ be the subset of $\mathbb{F}_{2}^{n}$ given by $k_{1} + k_{2} + \cdots + k_{n} = 0$. Prove that ...
48 views

### Solutions of $x^d=1$ in a finite field

Let's consider the polynomial $x^d-1$. Theory tells us that it can have at most $d$ roots in (any extension of) a given field. Here's my problem: let $A$ be the vector space spanned by ...
56 views

### Find a basis of E as a vector space over $\mathbb{Q}$

Find a basis for the factor ring $$\frac{\mathbb{Q}}{<16x^4-30x^3+15x^2+6>}$$ as a vector space over $\mathbb{Q}$. I honestly don't even know how to start this :( I though I would use ...
49 views

### Intuition for the fact that, in a vector space V over a field F, av = 0 $\implies$ a = 0 or v = 0. (a $\in$ F, v $\in$ V).

I have no trouble proving this: Let av = 0. If a = 0 then then we are done. Otherwise, there exists $a^{-1} \in F$ such that $a{^-1} a = 1$. Multiplying both sides of the equation by $a^{-1}$ gives ...
45 views

### Unique decomposition of a vector space into a direct sum

Suppose I have a vector space W that is the direct product of two subspaces, U and V. So: $W=U\oplus V$ My working definition of direct product is that $W = U + V$ and $U\cap V = 0$. Now my problem ...
52 views

### Left adjoint to direct sum?

In the category of vector spaces, is there some endofunctor $F$ satisfying $$\mathrm{Hom}_k(M,\underset{i \in I}{\bigoplus} k) \cong \mathrm{Hom}_k(F(M),k)$$ for every $k$-vector space $M$?
34 views

275 views

### Which are the most effective modern intuitive definitions of a vector?

First, I would like to clarify what I mean by "intuitive definition": an intuitive definition is an informal understanding of a concept which helps to build mental agility, with the possible ...
44 views

### Polynomial vector space terminology

Consider the vector space $P$ and the subset $V$ of $P$ consisting of those vectors (polynomials) $x$ for which a) $2x(0) = x(1)$, b) $x(t) = x (1-t)$ for all $t$. In which of these cases is $V$ a ...
128 views

### Are there any examples of vector spaces over non-numerical fields? If not, why not?

By non-numerical vector spaces I mean vector spaces that do not have as their scalars some sort of easily discernible numerical fields (e.g. complex numbers, functions are usually maps from one ...
68 views

### Can something like $\text{Hom}(V,K)$ be visualised?

I have no trouble visualising vector spaces like $\Bbb R^3$ and (e.g.) a subspace of dimension $2$, which would just be a plane through the origin of a $3$-D space, but I'm having trouble visualising ...
29 views

### difference between generate and base?

i know that when the dimension of a real vector space is equal to the number of vectors we have then they generate it. But they should be linearly independant so that they form a basis of this vector. ...
50 views

### When is a vector space (over field $K$) also a ring (with subring $K$)?

(Apologies in advance for the very naive question. I'm just learning about all this. Also, for the sake of expedience, below I use the word "ring" when it would more correct for me to use ...
83 views

### Matrix-free proof of $Z(GL_n(F)) = \{\lambda I:\lambda \in F^\times\}$?

How does one prove that $$Z(GL_n(F)) = \{\lambda I:\lambda \in F^\times\}$$ without resorting to matrices (and bases)? (BTW, $Z(GL_n(F))$ is the center of $GL_n(F)$, the general linear group of order ...
37 views

### Subspace of Division Algebra

I'm working on understanding the following proof: https://dl.dropboxusercontent.com/u/17606191/proof.gif but I'm having some trouble understanding some of the author's terminology. We're asked to ...
356 views

### In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their ...
148 views

### Orthogonal complement in a finite field ${\mathbb Z}^{n}_{q}$

When $V=\mathbb{Z}^n_q$ is a vector space, where $\mathbb Z_q$ is the set of integers modulo prime $q>2$, are the following statements true? If $U ⊂ V$ is a $k$-dimensional ...
29 views

### Dimension of ${U\cap W}$

I have the following question: Let $V$ be a vector spaces with dimension $n$. Let $U$ and $W$ be distinct sub vector spaces of $V$ with dimension $n-1$. Find the dimension of ${U\cap W}$. I proved ...
105 views

### Product of vector spaces

Let $V$ be a vector space over a fixed field $k$. Under what circumstances do we have $V\times V\cong V$? I think this should be true if $\mathrm{dim} \ V=\infty$, isn't it?
39 views

### dividing a line segment in the ratio $1:2i$

The following exercise is from [Birkhoff and MacLane, A Survey of Modern Algebra]: Let $\alpha=(1,i,0), \beta=(0,1-i,2i)$. Can you divide the line segment $\overline{\alpha \beta}$ in the ratio ...
100 views

### Complements of subspaces and quotient spaces

I could use a hint on the following question Exhibit vector spaces $A$, $B$, $C$, and $D$ such that $A \oplus B = C \oplus D$, $A \cong C$, but $B \not\cong D$. I have toyed around with a few ...
75 views

Let $V$ be the real vector space of 2x2 matrices and $End (V)$ the space of all linear transformations of V in V. $$T: V \rightarrow End (V)$$ $$T(A)(B)=AB-BA$$ I have to prove that this is a linear ...
61 views

### Complete Lattice, Complemented Lattice, Modular Lattice of Subspaces of a Vector Space

Let $V$ be a vector space over a division ring $D$ and $S$ the set of all subspaces on $V$, partially ordered by set theoretic inclusion. (i) $S$ is a complete lattice (ii) $S$ is a ...
74 views

### Dimension of Direct sum of same Vector Spaces

If $V$ is a finite dimensional vector space and $V^n$ is the vector space $$V\oplus V\oplus ...\oplus V\quad(\text{n summands})$$ then for each $n\geq 1$, $V^n$ is finite dimensional and dim ...
29 views

### Explicit formula for a right splitting once we have a left splitting

Assume we have a short exact sequence (of abelian groups or vector-spaces, it doesn't matter) $$0\rightarrow A\stackrel{\iota}\rightarrow B\stackrel{\pi}\rightarrow C\rightarrow 0.$$ If we have a ...
124 views

### Basis of Vector space $\Bbb C$ over rational numbers.

What will be the basis of vector space $\Bbb C$ over field of rational numbers? I think it will be an infinite basis! I think it will be $B=\{r_1+r_2i \mid r_1, r_2 \in \Bbb Q^{c}\}\cup\{1,i\}$. ...
35 views

### Span of a f.g. $R$-module over the quotient field of $R$.

Let $R$ be an integral domain with quotient field $K$, $R \neq K$. Let $V$ be a finite dimensional vector space over $K$ and $M$ a finitely generated $R$-submodule of $V$. My question is how $KM=V$ is ...
60 views

### Vector space $V$ over $\mathbb{Z}_p$ when $p$ is a prime

I need to determine if the following statement is true or false, if it's true, I need to prove it, else I need to give a counterexample: Let $V$ be a vector space over $\mathbb{Z}_p$ when $p$ is a ...
27 views

### $V=F_1^2$ over $F_2$: which operation to choose?

I don't understand the following vector space: $$V=\{(x,y)|x,y\in F_1\}$$ and $V$ is over field $F_2$, ($F_1$ is a field too). My question is: Is $V$ really a vector space? I am not talking about the ...
252 views

89 views

### Is a vector space a ring, integral domain or field?

Is a vector space a ring, integral domain or field, with respect to scalar multiplication? If you could give me an example, that would be awesome!
21 views

### A large set of low dimensional vectors in $\mathbb{F}_2^L$, which sums of any small subset do not cancel.

Fix a number $n$ and $L=O(\log n)$. Let $S=\{v_1,\dots,v_n\}$ be a set of vectors where $v_i\in \mathbb{F}_2^L$. We say that $S$ is "$\alpha$-good" iff for any nonempty subset $T \subset S$ where ...
### Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV
Let V be a real n-dimensional vector space. Show that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is isomorphic to V + iV. Note that V $\otimes _\mathbb{R}$ $\mathbb{C}$ is a real vector space and is ...