Question

. Let T : R n → R m be a linear transformation and A the standard matrix of T. (a) Let BN = {~v1, . . . , ~vr} be a basis for ker(T) (i.e. Null(A)). We extend BN to a basis for R n and denote it by B = {~v1, . . . , ~vr, ~ur+1, . . . , ~un}. Show the set BR = {T( r~u +1), . . . , T( ~un)} is a basis for range(T) (i.e. col(A)). Conclude that dim(col(A)) = n − dim(Null(A)). (b) Show that row(A) = Null(A) ⊥ (i.e. ker(T) ⊥). (c) We have established that for any subspace W of R n (or of any finite dimensional vector space V ), dim(W) + dim(W⊥) = dim(R n ) = n (or dim(W)+dim(W⊥) = dim(V )). Conclude that dim(col(A)) = dim(row(A)).

Answer 1