The purpose of this problem is to prove the Rank-Nullity Theorem.

Let T : Rn Rm be a linear map. Let {u1 ,...,uk} be a basis for ker(T), and extend it to a basis {u1 ,...,uk ,v1 ,...,vr } of Rn.

(a.) Why must r = n k?

(b.) Show that range(T) = span{T(v1 ),...,T(vr )}.

(c.) Show that T (v1 ), . . . , T (vr ) are linearly independent.

(d.) Conclude that

dim (ker (T )) + dim (range (T )) = n.

This is a linear algebra question. please solve and show all the necessary steps. write all the assumptions. and please use clear hand writing.

Thanks a lot.