How does one compute the number of ring homomorphisms

[Treadstone 71]

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?

 

[matt grime]

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.

 

[Treadstone 71]

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)?

 

[mathwonk]

make a conjecture and try to prove it.