Problem 9 Suppose V is a vector space with dim V = 3, having basis B =64], iii-'3} 6 8 4 Let T: V } V be a linear transformation and let the matrix of T with respect to B be A = 3 3 3] 4 E: 6 1 Let {E V such that lB = 3 (where t]B denotes the coordinate vector of v with 4- respect to B). (a) What is [T6r)]B? Now, let B'={i'1 + if3, {'2 + 2 iF3, 2 i'1 + 3?; 5L3}. You can assume that B' is a basis ofV. (b) Let S be the changeofbasis matrix 'om B' to E. Find S. (c) Use your answer to part (b) to find the changeofbasis matrix from B to 3'. Problem 10 In the vector space F (R), let B = {Exx Xex'XZ Ex} (you can assume this set is linearly independent) and let W = span B. Let D: FUR) > F (1R) be the linear transformation whose rule is D(f) =f' (the derivative of f)- (a) Prove that DUN) E W (that is, D applied to any vector in W is a vector in W) (b) Find the matrix of D with respect to B (c) Why can we conclude that D must be an invertible transformation