Suppose T: P^2--P^1 is a linear transformation whose matrix with respect to the bases B={1, 5-x, 2+3x-x^2} and B' ={x+3, 2} is given by: [T]_B,B' = [4, -1, 5; -7, 0, -2]. (a) Find _B. (b) Use (a) to compute T(6x^2-3x+8). Don't forget to decode! (c) Use the idea of (a) and (b) to compute T(1), T(x), and T(x^2). Computational Hint: You can find the three coordinate vectors at the same time by solving a 3 x 6 augmented matrix. (d) Use (c) to construct [T]_S,S', where S = {1, x, x^2} and S' = {1, x} are the standard bases for P^2 and P^1, respectively. (e) Use [T]_S,S' to recomput

Question

Suppose T: P^2-->P^1 is a linear transformation whose matrix with respect to the bases B={1, 5-x, 2+3x-x^2} and B' ={x+3, 2} is given by: [T]_B,B' = [4, -1, 5; -7, 0, -2].
(a) Find <6x^2-3x+8>_B.
(b) Use (a) to compute T(6x^2-3x+8). Don't forget to decode!
(c) Use the idea of (a) and (b) to compute T(1), T(x), and T(x^2). Computational Hint: You can find the three coordinate vectors at the same time by solving a 3 x 6 augmented matrix.
(d) Use (c) to construct [T]_S,S', where S = {1, x, x^2} and S' = {1, x} are the standard bases for P^2 and P^1, respectively.
(e) Use [T]_S,S' to recomput