Question

Problem 5. (a) Prove that for every ordered basis B = (61, ..., bn) of R”, there is a unique ordered basis B* = (1, ...,b*) of R™ such that 5.; = dij for all 1 Si, j <n.(Hint: think about matrix multiplication.] (b) Let E = (ēl, ..., ēn) be the standard basis of R”. Find E*. (c) Let B = (61, ..., 6n) be an ordered basis of R”, and let B* = (1, ...,5%) be as in part (a). ſoul Prove that for every vector ū e R", (ŪB = . (d) Let B = (61, ..., 5n) be an ordered basis of V, and let ŪEV. Can you express B** in terms of B? What is (ŪB? (e) Which ordered bases B = (61, ..., 5m) of R™ have the property that B* = B? 2Before continuing, make sure you understand what the vector addition and scalar multiplication operators are in L(V,V). Recall that "Sij" is the Kronecker delta defined by dij = 1 if i = j and dij = 0 if i #j.

Answer 1

Solution - Pooblem -5 (a) Take A= - 4 det(A) to since ( by bon) is a basis. == ايرادات 1 nome columns of t as B - le by - 3 (since B B & B o-os)

Since 22-1, e 3 = since t h e saib - {a C 2 ) = ai = 117112 [W] WAN

(e) Bt - B b + b - E, B + - 1,2,...h 116114-1 ie, ootto rosmal boes have the revised Property.