Question

Change of Bases 3. C (1-x, 1+x, x2} Consider the two sets of bases for P2. {1, 2 x, x + 2x2} B and Using the standard basis, 1, x, x2} for P2(x), express the vectors in B and C as B-coordinate and C- a) coordinate vectors relative the standard basis. Now show that B and C are independent sets. (3 pts.) Let vector u = 3 -5x +2x2. b) What is [u]B? (2 pts.) c) What is the change of bases matrix Pc-B? (2 pts.)

Answer 1

C - 1+* , x2> To vectors basis x,x33 expresS relative to standard 1 = | + O.x +0.x2 2 E 2-| + 1 . x + ox2 x 2x2 To expre ss vectors relative C to s tamalar basis 1 +(-1). x + o.x2 1 1 o.x2 2 O- ox + T,x2 To shouo vecto rs n linearly Cre independent b(2+ x) + C ( x+ 2 x2) Suppose a. 1 o Ca 2 b) -+ (b+<) + 20 x =O 4 0x +0x2 Сomparing coefficieuts a2b O 2C O C O a O tinearly inde pendent iv) To show linearly indepeudeut vectors n are a. 1-)+ b(1+2) suppose CA b O,_a +b = O

O a b a b 2 b a o b= o, C O C linearly independent. b u 3 -5 2 x2 2 +x) + Q3(x +2x3) a, ( a2 a,+ 292) + (q2+ a3)x + 293 x2 a, 29, 3 42aa 243 2 ai 3-2a2 a2 5- - -6 = 3+12 = 15 :. a1 15, z-6 15 - G bases matrix Chauge of c) - (a,a (- a+a2)x+ azx2- a 2 -a a2=o 292 a 42 a a 2 - a 42 = 1 a 2-3 242 3

t22 (a,ta2) C-a,t a2) -t 2 2 Cl2. C 2a2 =2 3 a =- PcB 2 2.