Math 304 Answers to Selected Problems1 Section 5.44. GivenA =1 2 21 0 23 1 1and B =−4 1 1−3 3 21 −2 −2Determine the value of the following(a) hA, Bi(b) kAkF(c) kBkF(d) kA + BkFAnswer: We use the inner product for R3×3given on page 245 of thetextbook.(a)hA, Bi = 1(−4) + 2(1) + 2(1)+1(−3) + 0(3) + 2(2)+3(1) + 1(−2) + 1(−2)= 0(b) kAkF=phA, AihA, Ai = 12+ 22+ 22+ 12+ 02+ 22+ 32+ 12+ 12= 25Thus, kAkF=√25 = 5.(c) kBkF=phB, BihB, Bi = (−4)2+12+12+(−3)2+32+22+12+(−2)2+(−2)2= 49Thus, kBkF=√49 = 7.1(d) kA + BkF=phA + BiA + B =−3 3 3−2 3 44 −1 −1hA + B, A + Bi = (−3)2+ 32+ 32+ (−2)2+ 32+ 42+ 42+ (−1)2+ (−1)2= 74Thus, kA + BkF=√74.7. In C[0, 1], with inner product defined by (3), compute(a) hex, e−xi(b) hx, sin πxi(c) hx2, x3iAnswer:(a)ex, e−x=Z10(ex)(e−x) dx =Z101 dx = 1(b)hx, sin πxi =Z10x sin πx dx=h−xπcos πxi10+Z101πcos πx=1π+1π2sin πx10=1π2(c)x2, x3=Z10x5dx=16x610=168. In C[0, 1] with inner product defined by (3), consider the vectors 1 andx.(a) Find the angle θ between 1 and x.(b) Determine the vector projection p of 1 onto x and verify that 1−pis orthogonal to p.(c) Compute k1 −pk, kpk, k1k and verify that the Pythagorean Lawholds.Answer:(a) The angle θ satisfiescos θ =h1, xik1kkxkWe need to compute h1, xi, k1k, and kxk:h1, xi =Z10x dx =12k1k =ph1, 1i =sZ101 dx = 1kxk =phx, xi =sZ10x2dx =1√3Thus,3cos θ =1/21/√3=√32Thus, θ =π6.(b) The vector projection of 1 onto x is determined by the formulap =h1, xihx, xixWe can compute h1, xi and hx, xi:h1, xi =Z10x dx =12hx, xi =Z10x2dx =13Thus, p =1/21/3x =32x.(c) We c ompute k1 − pk, kpk, and k1k:k1 − pk =ph1 − p, 1 − pi=s1 −3x2, 1 −3x2=sZ101 −3x22dx=vuut"13−231 −3x23#10=s−29−123+29=124kpk =php, pi=sZ1094x2dx=s34x310=√32k1k =ph1, 1i =sZ101 dx = 1Now, we can check the Pythagorean Law, which says thatk1 − pk2+ kpk2= k1k2We can check it:122+ √32!2=
View Full Document
Unlocking...