Determine the matrix forms of the following linear transformations with respect to the given bases. You may assume each of the following maps are linear. (a) Let V = P₂(R) and T: V → V be given by d T(p(x)) = = x + dx P(x). If α = {x + 1, x-1, x² + x} is a basis for V, find [T]%. (b) Let V = R³, W = R², and T : V → W be given by T(x1, x2, x3) = (x₁ + x2, 2x2 − x3). If a = {(1, 1, 0), (1, 0, 1), (0, 1, 1)} is a basis for R³ and ß = {(1, 1), (1,−1)} is a basis for R², find [7]. (c) Let V be the subspace of R4 spanned by {(1, 1, 0, 0), (0, 2, 1, 1)} and W = R¹. Let T: V→ W be given by the restriction to V of the map R4 → R¹; (X1, X2, X3, X4) → (X1, X2 — X3, X3 X4, X4 - X₁). - If a = {(1, 1, 0, 0), (0, 2, 1, 1)} is a basis for V and ß is the standard basis of W, find [T].
