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I want to count the number of linear maps $F_2^2 \to F_2^3$. I know that there are $3$ possible bases (assuming order is irrelevant) for the domain and $28$ possible bases for the codomain.

If I imagine that linear maps are analogous to functions in that the number of functions from $A \to B$ is $|B|^{|A|}$, then this answer/guess is given by $8^6$. I am wondering if that is the right way to go about it, in terms of linear maps, however. Would we have to use the bases in some way?

I read the similar question Counting the number of linear maps, but I am assuming I only know the basic definitions of linear maps and vector spaces - matrices have yet to be defined.

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    Curious: why avoid matrices? They're like the absolute best possible tool for problems like this. It's why they exist. – Randall Sep 23 '22 at 14:54
  • @Randall the reason is that this problem was presented in class, where we have not yet defined matrices, so I am wondering of a solution (even if is less elegant) that uses just the basic definitions –  Sep 23 '22 at 14:55
  • I just can't imagine a course involving linear maps between finite fields that wouldn't assume the student had already seen some linear algebra/matrices. I've been wrong before, though. – Randall Sep 23 '22 at 15:09

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I think I have found a solution at Number of linear maps between vector spaces over finite fields

Here, our field has $2$ elements, and we map from a space of dimension $2$ to dimension $3$. So there are $2^6 = 64$ possible linear maps.

It seems like the answer just uses the idea of matrices, but expanded out as linear combinations instead of directly using matrices and isomorphisms with matrices (so @Randall was right, matrices are the best solution).