Question

2. Let e1(x) = 1, ez(x) = x, p1(x) = 1 – x and p2(x) = –2 + x. Let E = (e1,e2) and B = (P1, P2). 2 a) Show that B is a basis for P1(R). 4 b) Let ce : R → R3 be the change of coordinates from E to ß. Find the matrix representation of C. Leave your answer as a single simplified matrix. 6 c) Let (:,:) be an inner product on P1(R). Suppose that ß is orthonormal with respect (:, :). Compute the following. Leave your answers as exact numbers. (Hint: Write each vector in terms of ß.] answers as i) ||P2 |= ? ii) (P1, P2) = ? iii) (e2, pı) = ? iv) ||e2| = ? v) (e2 + P1, P1) = ? vi) ||e2 + p1|| = ?

Answer 1

Let e, cx) = 1, ecc) = x, Pica) = 1-2, and Pe (= -2+. Let E = (en, es), and ß= (PiPe (a) Show that B is a basis for PJ CIR). solcetion: Pg URis a vector space whose dimension is 2. so any two linearly independent vectors will form a basis of Pz CIR ) so we will check whether pica) and P2 COX) are linearly independent or not. considera Pi («) + b P2(x) = oxto ~a (1-X) + b (-2+x) = 0x+o → (-a+b)x+ (9-26) = 0&+0 -a+b=0 solving for a and b we als and a-2b=0 we get a=o, b o 50 Pica and P2 (a) are linearly independent. Hence B = {Picae , Perovs} will be a basis of PeC1B) .. (b) Let CDIR→ IR2 be the change of coordinates from to B. Find the matrix representation of C. Leave your answer as a single simplified matrix, solution: Express Pica), Parve as linear combination of er«) and eg(x). so Pica) = ae, (X) + b eg x → 1-X = a (1) + 6(2) :,:. a=1 and b=-1 1-4 = (a+bre) also Pz (Q) = a, e, (X) + b, 22(X) al =-2 and b2=1 →-2+x = aril + base 1 1 (C) Let L., oy be an inner produeet on PA (IR). Suppose that Bu or thonormal with respect to <•,•). Compute the following

dor solution : Note that e, = a can be expressed as a line combination of pica and P2 (x) as.. eg (X) = a. pic«)+b.P2(X) x= a [1-6] +b (-2+x] = (-a+b) x + (9-26) 1-a+b= I solning for a fib we get a-2b = 0 . a -2, b= - ...e () = (-2) (1-2) + (-12.(-2+2) © 11 Pell = ? 11 Pall = { { pe , Pay}"2 = { <(1-4), (1-84373'? = (19'3 = (: P, CX) 4 P2 (6) are orthonormal w.rit.l.,)) :| P11 = 1 <P .1 P: Y = ? <P2, P,) = 4(1-2).(-2+X) = O( Pica and P2 (6) are orthonormal w.rit. {soy ) Pi, P2} = 0 fii Leg Pi> = ? Bering (eg, P,} = ((-2)(1-x)+(-1).(-2+), (1-x)} = (-2) <11-),(1-%D8+ (-1){(-2+x),(1-2) ; tee, pay = (-2) x1+ (-1) X0 → Lee, Pi> = -21 lle, 11 = ? lleg JI = { <ezien?}'2 ={ {(-2)(1-2) + (-1).(-2+2),(-2)C1=2)+(-12.(-2+29732 ={(-22 <61-), (-2)(1–)+(-1) (-2+4)* +(-1){(-2+x),(-2)(1-2) + (-1) (-2+x) > 1 2 =f(-2)(-2) 261-x)(1-X) (-2)(-1) 311-2).(-2+x)) +(-1) (-2) <(~2+), C1-2018 + (-106-10 11-2+x),(-2+297912 = { 4x1+2 x 0 +2 x 0 +1x13'2 = {4+0+0+13'2 Tell = 15e

(V) <ee+P1, P, y = ? <e+ P P, y = {(-2)(1-x)+(-1)(-2+2)+(1-), (1-0)* = *(-1) (1-2) + (-1) (-2+1), (1–2007 = (-1){(1-x),(1-2)}+(-1){(-2+), (ingby =(-1) X1 +1-1) XO >ke,+P , P, y = -1 (vi) 11g+Pill ? Now, e2+ P, = (-2)(1-2)+(-1)(-2+2)+(1-0) e, HP, = (-)(1-x)+(-)(-2;1) ... 11 +P, 1l ={{(-1(1-)+(-1) (-2+1),(-1)(1-2)+(-1)(-2+x)f/2 ={-1) {(1-2),(-1)(1-)+(-1)(-2+x) +(-1){(-2+), (-1) (1-7)+(-1)(-2+x)} } 2 54-1)(-1){(1-9) (1994+ (-1)(-1) <(1-9),(-2+x)} +(-1)(-1){(-2+4), (1-)}+(-1)(-1) 4(-2+2), (22+4)}}'z ={1x 1 + 1 x 0 + 1 x 0 + 1 x 142 = Illent P, V = V2