Name:Section: 9-10 11-12 2-3Math 54 Quiz 8 SOLUTIONSMarch 31, 2008GSI: Rob BayerYou have 20 minutes to complete this quiz. You must show your work.1. (2 pts)(a) Complete the following equation: Projuy =y·uu·uu(b) Define what it means for {v1, v2, . . . , vn} to be an orthonormal set of vectorsThere are many ways to answer this, but the shortest is probably: vi·vj=0 if i 6= j1 if i = j2. Let b1=1101, b2=−131−2, b3=−1011, y =433−1.(a) (2 pts) Show that {b1, b2, b3} is an orthogonal set1101·−131−2= −1 + 3 + 0 − 2 = 01101·−1011= −1 + 0 + 0 + 1 = 0−131−2·−1011= 1 + 0 + 1 − 2 = 0(b) (3 pts) Let W = Span {b1, b2, b3}. Write y asby + z whereby ∈ W, z ∈ W⊥Since the b0s are orthogonal, we can find by by calculating the projection onto eachvector:by =433−1·11011101·11011101+433−1·−131−2−131−2·−131−2−131−2+433−1·−1011−1011·−1011−1011= 21101+1015−131−2−23−1011=2400z = y −by =2−13−13. (3 pts) Let T : P2→ P3be the linear transformation defined by T (p) =Rt0p(x)dx. Find thematrix for T with respect to the bases {1, t, t2} and {1, t, t2, t3}(Hint: as an example, T (3 + 4t) =Rt0(3 + 4x)dx = 3x + 2x2|t0= 3t + 2t2− (0 + 0) = 3t + 2t2)T (1) =Zt01dx = tT (t) =Zt0xdx =t22T (t2) =Zt0x2dx =t33In {1, t, t2, t3}-coordinates, these are0100,00120,00013, respectively, so the matrix is:[T ] =0 0 01 0 001200
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