LetT:V->Wbe a linear transformation, and let the dimension ofVben, Let { } be a basis for the kernel ofT.

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!