# Size of subspaces of Fnp\mathbb{F}_p^n.

If Fq\mathbb{F}_q is the field of characteristic qq then Fnq\mathbb{F}_q^n is a vector space over Fnq\mathbb{F}_q^n of dimension nn.

It is apparently true that if V⊂FnqV\subset \mathbb{F}_q^n is a subspace of dimension mm then VV has exactly qmq^m elements but I can’t find a proof of this. It is easy to check if m=1m=1 or m=nm=n (: but even the case m=2m=2 is eluding me.

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