Let B denote the following list of elements of P3(R): v1 = 1, v2 = x + 1, v3 = x2, v4 = x3 ? x2. (a) Prove that B is a basis of P3(R). (b) Describe the dual basis of B. In particular, if ?1, ?2, ?3, ?4 are the elements of the dual basis, find ?i(a + bx + cx2 + dx3), 1 ? i ? 4. (c) Let S denote the standard basis ~e1, ~e2, ~e3 of R3. Let T : P3(R) ? R3 be given by T (a + bx + cx2 + dx3) = a~e1 + (b + c)~e2 + (c + d)~e3. Find the matrix of T with respect to the bases B and S. (d) Let S? and B? be the dual bases to S and B, respectively. Find the matrix of the dual map T ? : (R3)? ? P3(R)? with respect to these dual bases. (e) Find T ?(?)(5 + 3x + 2x3) where ? ? (R3)? is the linear functional ?(a~e1 + b~e2 + c~e3) = a ? 2b + c

' . Let 1?:- denote the f[)llUWlI'lg list [if elements Hf Pg}: Ti] 2 111,12 2 :r: + Leg 2 3221-114 2 3:3 :]'IE. (a) Prove that B is a basis of Pt]. {b} Describe the dual basis [if B. In particular, if :35. . (rig, tag, all are the elements at the dual basis, find ma. + tn: + (rs? + (13:3), 1 s: i. s: 4. {e} Let S denute the standard basis E 1,4532. a}; of R3. Let 1' : 'Pitt} > it\" be given by TU}. + tn: + (.2122 + {13:3} 2 as] + {h + ejr-Tg + [c + (Hf1:1. Find the matrix (If '1" with respect to the bases B and S. (d) Let 8' and B' be the dual bases in S and B, respectively. Find the matrix of the dual map I" : [Raf > Paw-ET with respect tu these dual bases. (e) Find T'{r,-'i}{5 + 33: +213] where qt: E [Raf is the linear functional {Mae} + as +c'3] = a: 2h + c