Showing linearity:

Choose 2 vectors in .

Choose

We want to show that :

Can you do that? It's just writing out.

Observe that ker . Hence v satisfies .

Find two independant vectors in that satisfy these relations and you have a basis for ker(T).

Find 2 more independant vectors: and you have a basis for Im(T)

(Dim(Im(T))+ Dim(ker(T)) = 4).