# Consider the field $F=Z_3$, with elements denoted [0],[1],[2], and the vector space $V=F^2$ over $F$, with elements denoted as pairs ([i],[j]).

Consider the field $$F=\mathbb Z_3$$, with elements denoted $$[0],[1],[2]$$, and the vector space $$V=F^2$$ over $$F$$, with elements denoted as pairs $$([i],[j])$$.

List all proper subspaces $$W\subset V$$, where `proper' means $$W\neq V$$. For example, you might describe the subspaces by listing all their elements. Give a brief justification of why your list is complete.

I'm confused about this question. How do I list all the proper subspaces? There are two many subsets there. Or did I just misunderstand the question?

• how is $F_2$ vector space over $F$ – user345777 Oct 3 '18 at 1:53
• @user345777 -- I'm sure $F^2$ is meant. – mr_e_man Oct 3 '18 at 3:13

$$V$$ is a two dimensional vectorspace over $$F$$. Therefore, all proper subspaces must have dimension strictly less than 2.
The zero dimensional subspace is trivially just $$\{([0],[0])\}$$.
The one dimensional subspaces are the subspaces spanned by a single vector. Since $$F$$ is finite, we can write all of these subspaces explicitly.
Let's start with the vector $$([0],[1])$$. We can compute the span of this vector since we are able to write every scalar in $$F$$. Therefore, the vectorspace spanned by $$([0],[1])$$ is, $$\{ [0]([0],[1]), [1]([0],[1]), [2]([0],[1]) \} = \{([0],[0]), ([0],[1]), ([0],[2]) \}$$
Now repeat this process with every vector in $$V$$. You will note that some of these spaces will turn out to be the same. For example, the space spanned by $$([0],[2])$$ is the same as that spanned by $$([0],[1])$$.