The kernel of T is, by definition, the set of all u in space U such that T(u)= 0. It is fairly easy to show that the kernel is a subspace. Choose a basis for the kernel and extend it to a basis for U. Call the subspace spanned by the extension U'. Then U= ker(T)+ U'. We can use that to show that any u in U can be written in a unique way as u1+ u2 where for u1 in the kernel and u2 is in U'. For any v in Im(T), there exist u in U such that T(u)= v. Write u= u1+ u2 where u1 and u2 are as before. Then T(u)= T(u1+ u2)= T(u1)+ T(u2)= T(u2)= v. Show that this is a "one-to-one" mapping of the image of T onto U'.