7. Consider the ordered bases B = (sin(3x), cos(3x), sin (3x), cos (3x), sin(3x) cos(3x)) and C = (1. cos(6z). sin(6z), cos(3z), sin(3z)) for a subspace W of C (R). (a) Find the transition matrix from B-coordinates to C-coordinates. (Hint: While this problem can be done using row reduction, it is easier, in- stead, to consider various trig identities that you learned in calculus.) (b) Use your answer to part (a) to express f(x) = sin(3x) (1 - 2 sin(3x) + 5 cos(3x)) in C-coordinates.

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Let B = {(1, 3), (-2,-2)} and B' = {(-12, 0), (-4, 4)} be bases for R², and let 23 = [33] 04 A = R2 relative to B. (a) Find the transition matrix P from B' to B. be the matrix for T: R² ->>> P = 6 9 [V] B [T(V)]B = (b) Use the matrices P and A to find [v] and [T(v)]B, where [V] B¹ = [-4 3]. -12 -1/3 -24 -96 4 -96 4 11 ← (c) Find P-1 and A' (the matrix for T relative to B'). 1/3

Let V be the space spanned by f1= sin x and f 2= cos x. (a) Show that gj = 2sin x + cos x and g2 3cos x form a basis for V. (b) Find the transition matrix from B' = {g1, g2} to B= {f1, f2}. (c) Find the transition matrix from B to B'. (d) Compute the coordinate vector [h] g, where h= 2sin x -Scos and obtain [h]B'. X> | (e) Check your work by computing [h]g' directly.