How does one compute the number of ring homomorphisms

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How does one compute the number of ring homomorphisms from \mathbb{Z}_2^n to \mathbb{Z}_2^m? Or, likewise, the number of linear mappings on those two vector spaces?
 
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By doing it. Pick a basis in one and work it out by hand. As vector spaces it's very easy, since it is just a finite set of matrices.
 
When asked the number of linear transformations between two vector spaces V and W, is it the same as asking the number of group homomoprhisms between V and W, of which are homogeneous? Is a group homomorphism automatically honogeneous (preserves scalar multiplication)?
 
make a conjecture and try to prove it.
 

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