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What is the process for finding all subspaces of a vector space over a finite field? Specifically, I want to find all the proper subspaces of the vector space $F^2$ over $\mathbb{Z}_3$. the $0$ vector space is the obvious subspace but I find that I have to guess a subspace then check all the criteria to determine if it is in fact a subspace. Also, is there a way to determine the number of subspaces that exist, apriori, or at least a maximum that can exist given the conditions?
Thanks in advance
Your example is a vector space of dimension 2, so the only proper subspaces are those of dimensions 0 and 1. You have already accounted for dimension 0. A vector space of dimension 1 consists of a single nonzero vector and all of its scalar multiples. So: pick a nonzero vector, gather all of its scalar multiples, there's a proper subspace. Then pick a vector not in that subspace, and repeat the exercise. Repeat until you have there are no more nonzero vectors left out, and you have all the proper subspaces.
The number of subspaces of a given dimension $k$ of an $n$-dimensional vector space over a finite field or order $q$ can be determined through a counting argument, in particular, by counting bases (count the number of sets of $k$ linearly independent vectors in an $n$-space, divide by the number of sets of $k$ linearly independent vectors in a $k$-space). The number of such subspaces is given by the Gaussian binomial coefficient $${n \brack k}_{q} = \prod_{i=1}^{k}\frac{q^{n-k+i}-1}{q^{i}-1}$$
(Note that if you wanted to count all of the subspaces, you would need to sum over $0 \leq k \leq n$)
