Define T by T: Pn → R by T(p) = =S² p(x) dx, where P is the set of all polynomials whose degree less than or equal to n. Find the n kernel of T.

Determine if the polynomials p(t) = 2 – t, q (t) = t2 + 2t are orthogonal under the inner product (flg) ,f(t)g(t)dt.

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.

Let A = 1 X x²x³ 1 X x²x³ x³ 1 1 X 22 8 Find such that det (A) = 0.