Lack of Basic Understanding about Linear Algebra/ Field Theory

Question

I came across the question:

One non-zero element of ($\mathbb{F}_2$)$^3$ is ($\bar{1}$, $\bar{0}$, $\bar{0}$). Write down three others which are non-zero and none of which are equal. Find a linear relation between these four elements. This proves the elements are linearly dependent. By quoting a statement (such as a Theorem, Lemma or Corollary), prove that four non-zero elements of ($\mathbb{F}_2$)$^3$ can never be linearly independent.

Foremost, I don't understand the notation: ($\mathbb{F}_2$)$^3$

Is this notation for a finite field of two elements in 3-dimensional space?

Secondly, I can't grasp the understanding needed to answer the question. Is the question just asking why a basis of an n-dimensional space should only consist of n elements? As opposed to, in this case, a 3-dimensional space basis can't contain 4 elements so they must be linearly dependent? Any help would be appreciated and sorry if this question is silly.

Answer 1

The notation $(\mathbb F_2)^3$ stands for $\mathbb F_2\times\mathbb F_2\times\mathbb F_2$.

Consider the elements $(1,0,0)$, $(0,1,0)$, $(0,0,1)$, and $(1,1,1)$ of $(\mathbb F_2)^3$. Then$$1\times(1,0,0)+1\times(0,1,0)+1\times(0,0,1)+1\times(1,1,1)=(0,0,0),$$which proves that they are linearly dependent.

Besides, for any field $F$ and any natural number $n$, a subset of $F^n$ with more than $n$ elements is always linearly dependent.

Answer 2

If $A$ is a set, $A^n$ is the set of $n$-tuples of elements from $A$. For instance, $\mathbb{R}^n$ or $\mathbb{C}^n$. If $\mathbb{F}$ is any field, $\mathbb{F}^n$ is a vector space over that field, where addition and scalar multiplication are performed in each coordinate, just as they are in $\mathbb{R}^n$ and $\mathbb{C}^n$. The notation $\mathbb{F}_2$ refers to the field with $2$ elements, or in other words the integers mod $2$. So $\mathbb{F}^2$ has $8$ elements $(a,b,c)$ with $a,b,c\in\{0,1\}$ chosen independently (viewing $0$ and $1$ not just as integers but integers mod $2$).

You should have some theorem available to you that if $X$ is a linearly independent set of vectors which span a vector space (i.e. a basis), then any set having any more than that will be linearly dependent. You can cite that theorem, alongside the observation that $\mathbb{F}_2^3$ is a vector space with a basis of size $3$ (the usual coordinate basis vectors), to conclude any set of $4$ elements must be linearly dependent.

Answer 3

$\Bbb F_2^3=\Bbb F_2×\Bbb F_2×\Bbb F_2$.

How about $(0,1,0),(0,0,1)$ and $(1,1,1)$ for your other vectors?

Then $(1,1,1)=(1,0,0)+(0,1,0)+(0,0,1)$.

$n+1$ vectors in an $n$-dimensional space are always linearly dependent.

Answer 4

Firstly, $\mathbb{F}_{2}$ is the finite field containing only two elements, $0$ (the identity for addition) and $1$ (the identity for multiplication). It is the smallest finite field there is. (see https://en.wikipedia.org/wiki/GF(2) for further explaination). $(\mathbb{F}_{2})^3$ is the $\mathbb{F}_{2}$ coordinate space in three dimensions, just like $\mathbb{R}^3$ is the real coordinate space in three dimensions.

An example of an element of $(\mathbb{F}_{2})^3$ would be $(1, 1, 0)$.

I am not sure how many lemmata or corrolaries you have already seen, but I think your suggestion in the post is the right one.