6. Find a basis of the null space of each of the differential operators below, and verify that the dimension of the null space is equal to the degree of the operator. (Use complex numbers where necessary for simplicity.) (a) D² - 4D +3 (b) D² + 2D + 2 (c) D³ (d) D (e) D¹ +4.

a) Use Newton interpolation to find the unique cubic polynomial f that passes through the points (0, 1), (1, 1), (2, 1), (3, 4). If we write f as a linear combination of binomial polynomials f(x) = E a, (:), what is the coefficient a2? %3D O 1 O -2 O -3

This is a question from a linear algebra course: Let V = R[X]3, the polynomials of degree at most three, and B = {1, X, X2, X3}. Show what the image under fB is of:• the four basic elements: P1(X) = 1, P2(X) = X, P3(X) = X2 and P4(X) = X3• P(X) = 2 + 6X + 3X2 + 4X3

There's a connection between the Existence and Uniqueness Theorem for constant-coefficient, linear, homogeneous IVP's and linear algebra.