Proving is a homomorphism is not hard. Proving 1-1 is even easier! So I dont understand why the textbook has done the mentioned manipulations...
To prove homomorphism: Let W = XY, the let us prove .
There is another way to show the map is onto. A consequence of the rank nullity theorem says that if you have a one-to-one linear transformation and the dimension of V and U are both equal and finite then the map is also onto. In this case which have the same finite dimension.