. Let T : R n ? R m be a linear transformation and A thestandard matrix of T. (a) Let BN = {~v1, . . . , ~vr} be a basisfor ker(T) (i.e. Null(A)). We extend BN to a basis for R n anddenote it by B = {~v1, . . . , ~vr, ~ur+1, . . . , ~un}. Show theset 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 haveestablished that for any subspace W of R n (or of any finitedimensional vector space V ), dim(W) + dim(W?) = dim(R n ) = n (ordim(W)+dim(W?) = dim(V )). Conclude that dim(col(A)) =dim(row(A)).
