Consider the field $F=\mathbb Z_3$, with elements denoted $,,$, 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?