Name:Section: 9-10 11-12 2-3Math 54 Quiz 5 SOLUTIONSMarch 3, 2008GSI: Rob BayerYou have 20 minutes to complete this quiz. You must show your work.1. (3 pts) Let B = {1 + t2, t + t2, 1 + 2t + t2} and C = {1 + 2 t, 3 + t2, 2 − 2t + 3t2}. Youmay assume that B is a basis for P2(a) Show that C is a basis for P2We have three vectors, namely120,301,2−23, in P2, which has dimen-sion 3 and thus we only need to show they are linearly independent:1 3 22 0 −20 1 3∼1 3 20 −6 −60 1 3∼1 3 20 1 10 0 2(b) Suppose p(t) is some polynomial such that [p]B=0−65. Find [p]CIf [p]B=0−65, then p(t) = 0(1 + t2) − 6(t +t2) + 5(1 +2t + t2) = 5+ 4t − t2. Sothen we just need to solve Ax =54−1, where A is the matrix from part (a):1 3 2 52 0 −2 40 1 3 −1∼1 3 2 50 −6 −6 −60 1 3 −1∼1 3 2 50 1 1 10 0 2 −2So we can back substitute to get x3= −1, so x2= 2 and x1= 1. Thus, [p]C=12−12. (3 pts) Find a basis for the subspace of R3spanned by12−1,24−2,274,11−31 2 2 12 4 7 1−1 −2 4 −3∼1 2 2 10 0 3 −10 0 6 −2∼1 2 2 10 0 3 −10 0 0 0So the pivot columns are 1 and 3 and thus our basis is12−1,2743. (4 pts) Let T : V → W be a 1-1 linear transformation from the vector space V to thevector space W . Prove that if v1, v2, · · · vpare vectors in V such that {T (v1), T (v2), · · · T (vp)}are linearly dependent, then {v1, v2, · · · vp} are linearly dependent.If {T (v1), T (v2), · · · T (vp)} are linearly dependent, then there are constants c1, c2, . . . cp,not all of which are 0, such thatc1T (v1) + c2T (v2) + · · · cpT (vp) = 0WSince T is linear, this is the same asT (c1v1+ c2v2+ · · · cpvp) = 0WSince T is 1-1, we must havec1v1+ c2v2+ · · · cpvp= 0VSince we chose the ci’s such that at least 1 is non-zero, we have shown that {v1, v2, · · · vp}are linearly
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