Let B = {b, b, b3} be a basis for vector space V. Let T: V → V be a linear transformation with the following properties. T(b,) = - 8b, - 6b2: T(b2) = - 2b, +4b2: T(b3) = - 7b, %3D Find [T]B, the matrix for T relative to B.

Let B= (b,, b,, b,} and D= (d,, d,} be bases for vector spaces V and W, respectively. Let T :V- W be a linear transformation with the following properties. T(b,) = - 3d, +2d2, T(b2) = - 4d, + 4d2, T(b3) = 5d2 Find the matrix for T relative to B and D. The matrix for T relative to B and D isO.

Let T : R³ → R³ be the linear transformation that does the following things, in this order, to an input vector x = [x y z]¹: (i) Interchanges the second and third coordinates of . (ii) Multiplies the first coordinate of the resulting vector by 2. (iii) Replaces the second coordinate of the resulting vector with a 0. (iv) Multiplies the resulting vector by the following matrix: 0 0 0 0 You don't have to show that T is linear. (a) The description of T given above is purely algebraic, in that it explicitly describes how to take 7 = [x y z] and write down T() in coordinates. Give a geometric description of what each of the four "steps" of applying T actually does to a vector. (Your Week 9 tutorials may help in describing what the last step does.) (b) Find the standard matrix AT of T. (c) Find a spanning set for null(AT), and describe what null(AT) is geometrically (i.e., describe it geomet- rically as a subset of R³) (d) Find a spanning set for im(AT), and describe what im(AȚ) is…