How Do Proofs for Vector Spaces Over Finite Fields Work?

In summary, Theorem 1 states that a vector space over GF(q) has \frac{1}{k!} \prod^{k-1}_{i=0} (q^{k}-q^{i}) different bases. Theorem 2 states that if S is a subset of F^{n}_{q}, then we have dim(<S>)+dim(S^{\bot})=n. The most important point is that the intersecton of C and C-perp may not be zero.
  • #1
jOc3
6
0
It's hard to find the proofs of these theorems. Please help me... Thanks!

Theorem 1: Let V be a vector space over GF(q). If dim(V)=k, then V has [tex]\frac{1}{k!}[/tex] [tex]\prod^{k-1}_{i=0}[/tex] (q[tex]^{k}[/tex]-q[tex]^{i}[/tex]) different bases.

Theorem 2: Let S be a subset of F[tex]^{n}_{q}[/tex], then we have dim(<S>)+dim(S[tex]^{\bot}[/tex])=n.
 
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  • #2
think about the steps involved in choosing a basis. then the numbet of bases is the number of ways to carry out these steps.

first choose any non zero vector. how many ways?

then afterwrads chooise avector that is not on the line through that one - hiw many ways?...
 
  • #3
What is F[itex]^{n}_{q}[/itex]? An n-dimensional innerproduct space over GF(q)? And is <S> the span of S?

My thoughts: get an orthonormal basis for <S>, and extend this to one for F[itex]^{n}_{q}[/itex]. The new vectors you add will probably be an o.n. basis for S[itex]^{\perp}[/itex].
 
  • #4
thm 2 is true in any vector space of finite dimn. over any field.
 
  • #5
morphism said:
My thoughts: get an orthonormal basis for <S>

This doesn't make sense over a finite field (or any field of positive characteristic).
 
  • #6
matt grime said:
This doesn't make sense over a finite field (or any field of positive characteristic).
I was wondering about this also. What is S[itex]^\perp[/itex] over a finite field?
 
  • #7
Just pick any complementary subspace, or take the quotient space.
 
  • #8
Sperp is the subspace of the dual space that annihilates S. equivalently, it is the dual space of the quotient by <S>, which is why matt's hint gives the right dimension.
 
  • #9
Ah, my bad. I wasn't familiar with that notation for the annihilator of S. I thought it was the space of vectors orthogonal to S.
 
  • #10
I need the proof for theorem2,too.

I know that theorem2 is an important property in coding theory.

(Fq)^n is the direct product over (Fq)^n (similar with R^n over R)


the S-perp in (Fq)^n have the same definition.

But the defition of inner space (over finite field) may not need <x,x> > 0.
I think the most important point is that the intersecton of C and C-perp may not be zero.
(i.e. it is possible that <x,x>=0 for some x =/= 0 in (Fq)^n)

Actually, in the language of coding. If C is a subspace of (Fq)^n with dim(C)=k,
C-perp = {x in C| x(G)' = 0 in (Fq)^k} where G is the generator matrix,(G)' : transpose of G
but I don't which is useful or not!:)
 

Related to How Do Proofs for Vector Spaces Over Finite Fields Work?

1. What is a vector space over a finite field?

A vector space over a finite field is a mathematical structure consisting of a set of elements, called vectors, which can be combined using operations of addition and scalar multiplication. The field in which the vectors are defined is finite, meaning it has a finite number of elements.

2. What is the importance of studying vector spaces over finite fields?

Vector spaces over finite fields have numerous applications in various fields of science and engineering, such as coding theory, cryptography, and error-correcting codes. They also provide a powerful tool for studying abstract algebraic structures.

3. How are vector spaces over finite fields different from vector spaces over real or complex numbers?

The main difference is that finite fields have a finite number of elements, while real and complex numbers have an infinite number of elements. This difference affects the properties and operations of vector spaces, such as the dimension and the existence of inverses.

4. What are some common examples of vector spaces over finite fields?

Some common examples include spaces of polynomials over finite fields, spaces of matrices with entries from finite fields, and spaces of n-tuples of elements from finite fields.

5. How are proofs involving vector spaces over finite fields different from other mathematical proofs?

Proofs involving vector spaces over finite fields often require a combination of abstract algebraic reasoning and concrete examples. They may also involve techniques from linear algebra and number theory. Additionally, the finite nature of the field introduces certain limitations and considerations that may not exist in other mathematical proofs.

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