Find a basis for the following subset of P2: { p(t): p'(1) = p(2) }

Define T: P2 → R by T(p) = S, p(x)dx. Find the kernel of T.

(a) A linear map is defined as T: P3 → P3 → T(p(x)) = xp'(x) - p(x), where p'(x) denotes the derivative of P(x), with respect to x. Find the basis for the ker (T).

4. Consider the function f(x) = (a¹x)(b¹x) where a, b, and x are n-dimensional vectors. a. Find Vf(x). b. Find the Hessian F(x).

(i) Consider T: R³ R³ defined by T(T1, T2, T3) = (x₁, I₁+I₂, X₁ + x₂ + x3). If B = {₁, ₂, 3} and B' = {B₁, B2, B3}, where a₁ = (0, 1, 1), a₂ = (1,0,1), a3 = (1, 1,0), B₁ = (1,0,0), B₂ = (1,1,0), B3 = (1, 1, 1), what is the matrix [T]B-B?