Let T : R^n rightarrow R^m be a linear transformation. Given a subspace Z of R^m, let 5 {v elementof R^n | T(v) elementof Z). Show that S is a subspace of R^n. Solution Let v,w in S Then T(v) and T(w) belongs to Z Consider av+bw , a,b are scalars We need to see if av+bw are in S For this consider T(av+bw) = aT(v)+bT(w) (because T is linear transformtaion) Now T(v) and T(w) are in Z And so aT(v) and bT(w) are in Z This => aT(v)+bT(w) are in Z Hence we get that T(av+bw) is in Z So we get av+bw is in S Hence S is subspace of Rn.