Consider the following bases B and B' for some subspace of M2x2(R), the vector space of all 2 by 2 matrices over R. u- {[: :] [:}] [:}} - -{[: :] [: :} [:)]} {[: :]· [: :]. [: :]}- {{: :]· [: :]· [: ']} 0 0 3 4 1 0 1 1 3 1 3 0 3 B B' 2 3 4 3 2 1 0 -8 -6 -1 (a) Find the coordinate matrix of A = 4 relative to the basis B', that is 6 [x(A)]g. (b) Calculate the transition matrix PB→B. (Show steps.] (c) Use part 2(b) to calculate the coordinate matrix of A relative to the basis B. (d) Verify the coordinate matrix obtained in part 2(c).

The functions 1, cos x, sin x, cos 2x, sin 2x, ... are a basis for Hilbert space. Write the derivatives of those first five functions as combinations of the same five functions. What is the 5 by 5 "differentiation matrix" for these functions?

(1) Consider the basis 8-{(:) (:)} B = of R2. What is the coordinate vector of (;) with respect to B.

3: Which of the following sets are closed under scalar multiplication? (1) The set of all vectors in R of the form (a, b, 9ab). (i1) The set of all polynomials a- bx+ cx in P, such that ac =D0. (iii) The set of all matrices in Mn such that A = A (A) (1) and (ii) only (B) (1) and (iii) only (C) (ii) and (iii) only (D) (1) only (E) (ii) only (F) none of them (G) (iii) only (H) all of them