Question

Problem 2. Recall that for any subspace V of R", the orthogonal projection onto V is the map projy : RM → Rn given by projy() = il for all i ER", where Ill is the unique element in V such that i-le Vt. For any vector space W, a linear transformation T: W W is called a projection if ToT=T. In each of (a) - (d) below, determine whether the given statement regarding projections is true or false, and provide a proof or a counterexample accordingly. (a) For every subspace V of R", the orthogonal projection projy:R" + R" is a projection. (b) For every projection T:R" +R" and basis B of R", the B-matrix (T B of T is a projection matriz, meaning that (T]B)2 = (T] (c) For every projection T:R" + R", we have T = projy where V = im(T). (d) For every vector space W and projection T: W W , if T is surjective then T is the identity map on W.

Answer 1

Problem 2. True; basis Let u be any subspace of R and let ß- & vig... } iis orthogonal of V, where k- den ) . Then, projv: R R is defined as proj, (*) - Žviri " ial Vigriz iz , in Cyg - ) ( ( ) ( و i v Trigri vi a proj v(a) Z Theatre. Truel Let I be any het ß be basis projection of IR? from Br tollen

Additional Sheet Let of Ik". B=(.. – Vas be basis for i = l, - gn. E e;- ith dedicann of meetrix (772 Telia ith column of meetrix (TDB T2 (vi) = T) for i=1, - en (7²)] = vi for 1-1, n. ( [vi] = (7] VIR for i=1) – en (7) [vi] = [], [hidrs for it, wh ith column of matox (7) Trei = (TVI, 9th elan each dumn of matorx TB matux (72 o. of © Let T: 1Rk v=Im (T) 18? = I'm (T) OW be a Topone for projection To Proju some subspan

TOT = 7² _T Let UER Uz utw ter for some VEV and tot W V- Pre) . Tv) T (TOT) (W)- T(U) E Im G) T(U)= v ker some vel I (Tea) - Tu - TW T (u) = vt T (T(4) - Tu=0 T (Tu)-u)=0 Tas-u & Kernelft) =W Tas-u & W Tw-u w for some w' ² T (a)= utw' = utw to' A n π T T π 1 V = TU) = V + w+Lol v'-v = wtw & Vow = {0} ul-v=0 v=ul : Tu)= v' = v= prejv (4) Ta projv Time Let woow be any element Since, T is surjective, there exich NEW such that Tw=u T² (u) = T (0) Te) = 7² (0) - T = T (U-w)= Tuj-T@) = 0 (d

- w-wt kernelf By sunk nullity then dem Imal) + dim (kernela)) = dim (W) dem Chernel I)) = dim (W) - dim (Im (A)) din (W) - dim (W) = 0 kernel (T) = {0} = w :: T (W) = T(U)= w - T is identity map