5) Let V be 3 dimensional with basis 3 = (81,82,193), and let go : V > V be a linear map (a) (3 points) Prove that there exists a linear map Tr(:p) : A3(V) r A30?) that sends the simple tensor Tr(tp)(el A 82 A 83) 2 99021) A e; A e3 + e] A 99032) A 63 +191 A62 A @033) We call this map the trace map of (p, and we call the resulting number that scales the basis vector 81 A 82 A 83 the trace of (p (b) (3 points) Consider the case V : R3 with standard basis. View the 3x3 matrix (111 012 {113 A: (121 022 023 HER\"Rs 031 032 033 as a linear map and verify that our denition of trace lines up with the usual one