0
votes
1answer
13 views

Prove the number of unordered pairs of linearly independent elements

Let $V$ be a vector space over $K$. Let $K={\mathbb{Z}}/{p\mathbb{Z}}$, and $\dim V=3$. We know that $V$ has $p^3$ elements. I need to show that the number of unordered pairs of linearly ...
2
votes
0answers
24 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 ...
1
vote
2answers
49 views

Weird field notation

I have a question: Let $\mathbb{F}$ be any field characteristic $0$. Recall that $x_i$, denotes the $i^{th}$ entry of a vector $x\in\mathbb{F}^n$. Define $$S = \{x\in\mathbb{F}^5 \mid x_i = ...
0
votes
1answer
20 views

fields and subspaces

Let F be a field and let V=F^F, which is a vector space over F. Let w be the set of all functions f element of V satisfying f(1)=f(-1). Is W a subspace of V? a. Has the zero vector b. closed under ...
1
vote
1answer
29 views

“$q$-linear envelopes” of $\mathbb{F}_p$-subspaces

Let $V$ be a vector space over an algebraically closed field $k$ of characteristic $p>0$, and denote by $V_q$ the vector space obtained from $V$ by restricting scalars to $\mathbb{F}_q$, where ...
0
votes
1answer
43 views

find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
1
vote
0answers
34 views

permutation polynomial

If we have GF(4) as an extension field, we can define a permutation polynomial of GF(4) like L(x), a linearized polynomial, of the followinf form: L(x)= \sum_{s=0}^{\r-1} a_s x^(q^r)e Is it possible ...
0
votes
1answer
52 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 ...
0
votes
2answers
24 views

Can a set containing a single vector from a vector space over a finite field be linearly dependent?

Take the set $S=\{v=(1,1)\}\subset F_2 ^2$. $v+v=(0,0)$ is a linear combination of vectors from $S$. Is $S$ linearly dependent?
1
vote
1answer
41 views

generating set for finite fields

let us consider GF(2^n) as a vector space over GF(2), Is it possible to find a generating set for GF(2^n)? How can ew find it? I want to define a linear transformation of GF(2^n) to itsefl.
0
votes
1answer
73 views

elementary abelian groups and finite fields

in group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order $p$, where $p$ is a prime; it is a particular kind of $p$-group. now suppose that we ...
3
votes
1answer
61 views

automorphism of fields

let we consider $GF(p^n)$ as a vector space over $GF(p)$, $p$ is prime. Also we want to have an invertible linear map on $GF(p^n)$, (automorphism of $GF(p^n)$). on the other hand we know that A field ...
3
votes
2answers
44 views

Is vectorspace trivial under these conditions?

Let $R$ be a ring. Looking for a left $R$-module free over abelian group $A$, I arrived at $\left|R\right|\otimes A$ with $r.\left(s\otimes a\right)=rs\otimes a$ where $\left|R\right|$ denotes the ...
2
votes
1answer
62 views

tensor product of a vector space and finite field

I want to know how we can interpret and define the tensor product of V as a vector space with a F as a finite field?
0
votes
1answer
66 views

Finding maximal number of bad triplets

Let $a,b,c\in \mathbb{F}_{3^n}$. The summation of two vectors is done with modulo $3$. The elements of vectors are $0,1$ or $2$. We will say that $a,b,c$ form a bad triplet if $a\neq b,a\neq c,b\neq ...
2
votes
1answer
85 views

Basic concepts in finite fields

I need some help with clearing up some some basic concepts in finite fields. I understand that $\mathbb{F}_p = GF(p)$ where $p$ is a prime is a finite field, which is isomorphic to ...
16
votes
1answer
211 views

Combinatorics in finite vector space

Let $q$ be a prime power and $V$ a finite $\mathbb F_q$-vector space with two subspaces $I$ and $J$. Let $k$, $a$ and $b$ be non-negative integers. Determine the number of subspaces $K$ of $V$ ...
0
votes
1answer
47 views

Lin Alg- Dual Spaces

Let $(V^*)^*=V^{**}$. Define $S:V\to V^{**}$ by $s(v)(\alpha)=\alpha(v)$ for all $v\in V$ and $\alpha\in V^*$. I need to show that $s(v)\in V^{**}$. And show the S is a linear transformation. ($V$ is ...
11
votes
1answer
155 views

Orientation on finite dimensional vector spaces over finite fields.

For finite-dimensional $\mathbb R$-vector spaces, we define an orientation to be an equivalence class of ordered bases, where $B_1 \sim B_2$ iff the change of basis matrix $A$ taking $B_{1}$ to ...
4
votes
1answer
53 views

number of 1-to-1 linear functions on vectorspaces over finite fields

This is not a homework. I just ask this question myself and thought it would be easy to figure out. But I did not get the solution. Let $\mathbb{F}$ be a finite field with $|\mathbb{F}|=q$. Consider ...
-1
votes
3answers
169 views

Possible Cardinality of a Field

The following question struck me as pretty interesting: Let $\Bbb F$ be a field of characteristic $p$ (a prime, of course). I'm then asked to show that $|\mathbb{F}| = p^n$ for some $n\geq 1$. ...
3
votes
1answer
490 views

Do there exist vector spaces over a finite field that have a dot product?

I've stumbled in the german wikipedia over the question if a vector space over a finite field exists, that has a dot product. Definition of dot product A dot product over a $\mathbb{K}$-vector space ...
2
votes
1answer
273 views

Linearly dependent vectors over finite fields

My problem is as follows: Assume you have a vector space of dimension $(d + 1)$, with values over $GF(q)$. Every vector in this vector space can be regarded as an element of the extension field ...
1
vote
1answer
372 views

The number of subspaces of a vector space

Let $V$ be a vector space of dimension $n$ over $\mathbb{F}_q$, and let $U$ be a subspace of dimension $k$. I want to compute the number of subspaces $W$ of $V$ of dimension $m$ such that $W\cap U=0$. ...
15
votes
3answers
1k views

Inner Product Spaces over Finite Fields

Inner product spaces are defined over a field $\mathbb{F}$ which is either $\mathbb{R}$ or $\mathbb{C}$. I want to know what happens if we try to define them over some finite field. Here's an ...
5
votes
2answers
1k views

Finite fields as vector spaces

I'm having great difficulty understanding this topic. Can someone concretely explain what it is meant by thinking of $GF(q^2)$ ($q$ a prime power) as a two-dimensional vector space over its subfield ...
1
vote
1answer
311 views

Numbers of vectors in a vector space over a finite field, with different multiplication

I had a recent question in an assignment that I couldn't complete. We are given the following: $q$ is an odd prime power. $(F,+,\cdot)=\text{GF}\left(q^2\right)$. $K$ is the $q$ element subfield of ...