[1] 5 Let u₁ = , նշ = 5 , U3 = 2 (a) Find the transition matrix T corresponding to the change of basis from {e1, e2, e3} to {u₁, U2, U3} T= (b) Find the coordinates of the vector x = -2 with respect to the basis {u1, U2, u3} [X]U

(a) Determine the transition matrix corresponding to a change of basis from the standard basis {e1, e2} to the ordered basis {u1, u2}. Use this transition matrix to find the coordinates of x = (1, 1)T with respect to {u1, u2}. (b) Determine the transition matrix corresponding to a change of basis from the ordered basis {v1, v2} to the ordered basis {u1, u2}. Use this transition matrix to find the coordinates of z = 2v1 + 3v2 with respect to {u1, u2}.

Find the matrix A' for T relative to the basis B'. T: R² A' = X A' = 3 11/3 R², T(x, y) = (x - y, y - 2x), B' = {(1, -2), (0, 3)} -3 48 -16 -4 Find the matrix A' for T relative to the basis B'. ↓ 1 T: R² → R², T(x, y) = (-7x + y, 7x - y), B' = {(1, −1), (-1,5)} -72 24

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.