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?

  • $\begingroup$ how is $F_2$ vector space over $F$ $\endgroup$ – user345777 Oct 3 '18 at 1:53
  • $\begingroup$ @user345777 -- I'm sure $F^2$ is meant. $\endgroup$ – 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 $([0],[1])$.

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