Let T: V -> W be a linear transformation, and let the dimension of V be n, Let { } be a basis for the kernel of T.
1) Let V_(m+1), v_(m+2), ..., be vectors in V such that { , , ..., , v_(m+1), ..., } in a basis for V. Prove that {Tv_(m+1), Tv_(m+2), ..., } is a basis for the image of T.
2) Using 1), prove that the rank of T plus the nullity of T is n.
Thanks!