# Thread: Vector spaces and homomorphism.

1. ## Vector spaces and homomorphism.

1) If V and W are finite dimensional vector spaces over F with
dimension m and n respectively, then prove that Hom(V,W) is also a
finite dimensional vector with dimension mn.

2) If dim V=m, then pt dim Hom(V,V)=m^2

3) If dim V=m, then pt dim Hom(V,F)=m

(Hom --> Homomorphism)

where Hom(V,W)={ T : V --> W, where T is homomorphism }.

2. Hi there!!!

Let b1,...,bm be a basis for V and let c1,...,cn be a basis for W.
One way is this:
define m times n linear maps f_{i,j}: V to W like this:
f_{i,j} (b_k) = cj if i=k and 0 otherwise.
Then one can show that the f_{i,j} are linearly independent in HOM(V,W),
and that each linear map in HOM(V,W) can be written as linear combination
of the f_{i,j}. That's pretty much it (you need to work out the details a bit).
The other two statements follow as a corollary
(recall that F is a one dimensional
vector space over itself).

Best,

ZD