4
votes
1answer
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 ...
2
votes
1answer
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 ...
0
votes
1answer
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 $k(x)$?
3
votes
1answer
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 ...
0
votes
3answers
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 ...
5
votes
0answers
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 ...
3
votes
1answer
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 ...
0
votes
1answer
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 ...
1
vote
1answer
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 ...
1
vote
1answer
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 ...
1
vote
1answer
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 ...
4
votes
1answer
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 $M$?
2
votes
2answers
34 views

Direct product commutes with coproducts?

Do direct products commute with the direct sums of vector spaces? Basically is $\underset{i \in I}{\prod} \underset{j \in J}{\bigoplus}M_{i,j} \cong \underset{j \in J}{\bigoplus}\underset{i \in ...
3
votes
1answer
54 views

“Vector spaces” over a skew-field are free?

Are modules over a skew field free? That is, if $F$ is a skewfield then can any module $M$ be written as $\underset{i \in I}{\bigoplus} F$ for some indexing set $I$?
2
votes
3answers
232 views

Finding a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$.

I have to find a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$. I determined that $\sqrt{2}+\sqrt{3}$ satisfies the equation $(x^2-5)^2-24$ in $\Bbb{Q}$. Hence, the basis should be ...
1
vote
3answers
155 views

The kernel and image of $T^n$

I need help with this question: Let $V$ be a finite vector space where $ \dim V = n $, over the complex numbers and let $ T: V\to V $ be a linear transformation. Prove that $ V = \ker(T^n) \oplus ...
3
votes
6answers
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 ...
1
vote
2answers
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 ...
2
votes
3answers
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 ...
5
votes
1answer
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 ...
0
votes
2answers
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. ...
0
votes
0answers
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 ...
2
votes
3answers
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 ...
0
votes
1answer
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 ...
29
votes
1answer
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 ...
7
votes
3answers
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 ...
0
votes
1answer
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 ...
3
votes
1answer
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?
2
votes
1answer
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 ...
0
votes
2answers
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 ...
2
votes
2answers
75 views

About linear transformations

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 ...
1
vote
1answer
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 ...
2
votes
1answer
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 ...
2
votes
1answer
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 ...
1
vote
3answers
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 $B=\{r_1+r_2i \mid r_1, r_2 \in \Bbb Q^{c}\}\cup\{1,i\}$. ...
2
votes
1answer
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 ...
3
votes
1answer
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 ...
0
votes
1answer
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 ...
0
votes
1answer
252 views

Dimension of direct sum of vector spaces [duplicate]

Let $V$ and $W$ be finite dimensional vector spaces on a field $F$. Show that $\dim(V\oplus W) = \dim V +\dim W$. My idea: let $\dim V=n$ and $\dim W=n$. So $\mathcal{A}=$ {${v_1 , v_2 ,... , ...
3
votes
1answer
71 views

Has anyone succeeded in formalizing the notion of a complete vector space? (Not using topological ideas).

In order theory, we have the concept of a lattice, which is defined as consisting of an underlying set $L$ together with two binary operations $\wedge$ and $\vee$. Now when $L$ is finite, the concept ...
1
vote
0answers
52 views

What vector space is this?

Let $a,b,c$ be odd primes. In particular, $ab, ac, bc$ are all odd numbers. We can use this to our advantage, since then $\sqrt[ac]{x} : \Bbb{R} \to \Bbb{R}$ is well-defined and a bijection. Let ...
1
vote
3answers
98 views

what is the relationship between vector spaces and rings?

Can you show me an example to show how vector and scalar multiplication works with rings would be really helpful.
2
votes
0answers
69 views

Recognizing pure tensors in tensor product of vector spaces

Let $V$ be a vector space and let $\{e_i\}$ be a basis for it. Then $\{e_I\equiv e_{i_1}\otimes...\otimes e_{i_r}\}$ is a basis for $V\otimes ... \otimes V$. Suppose I am given an element $w=\sum a_I ...
1
vote
2answers
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!
2
votes
0answers
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 ...
2
votes
2answers
170 views

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 ...
2
votes
1answer
65 views

Question about exact sequences and vector spaces

I am having much trouble trying to understand the idea of an exact sequence. As a toy example, I'm looking at the following exact sequence: $$0 \xrightarrow{} ker(T) \xrightarrow{\iota} V ...
0
votes
1answer
40 views

Endomorphisms of vector space

Let $E$ be a vector space of dimension $n$. Find all endomorphisms $f$ of $E$ which satisfy $f\circ f = \operatorname{Id}_E$ Is trivial that $f = \operatorname{Id}_E$ is a solution, but I don't ...
0
votes
1answer
75 views

Extension of a linear map to a commutative graded algebra

Let's fix the notation, $V=\bigoplus_{i\geq 0}{V^i}$ is a graded vector space and $\Lambda V$ is the free commutative graded algebra on $V$. I have been struggling to understand this example: ...
0
votes
0answers
42 views

Extension of multiplication to the tensor algebra.

In this wikipedia article http://en.wikipedia.org/wiki/Tensor_algebra#Construction We construct $T(V)$ as the direct sum of vector spaces $T^kV$ for $k=0,1,2,…$ $$ T(V)= ...