Recall that M2,2 is the vector space of real 2 x 2-matrices and P3 is the vector space of real polynomials of degree at most 3. Let T : M₂,2 P3 be a linear function satisfying Find the following: ( (a) T (b) T (c) T (d) T (e) T 1 0 0 1 C −1 1 = || || → T ([1]) = T + ( [i !]) = = -4, T([:]) = -4x², T = -x - 3, ([i]). = = 3x³ - 4x.

Let P3[x] be the vector space of all polynomial of order at most 3. Suppose that B = { 1, (x-1), (x-1)2, (x-1)3}. Show that B is a basis for P3[x]. Give p(x) = x3+ x, find [p(x)]B.

3: Which of the following sets are closed under scalar multiplication? (1) The set of all vectors in R of the form (a, b, 9ab). (i1) The set of all polynomials a- bx+ cx in P, such that ac =D0. (iii) The set of all matrices in Mn such that A = A (A) (1) and (ii) only (B) (1) and (iii) only (C) (ii) and (iii) only (D) (1) only (E) (ii) only (F) none of them (G) (iii) only (H) all of them

Let d : P5(R) → P5(R) denotes the first derivative. Show that d is a linear map. Choose a basis for P5(R) and find the matrix M(d). Use the matrix M(d) to find the derivative of f(x) = −9x^5 + 11x^4 −4x^2 +5x−1.