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Let V = {p element of R[x] | deg(p) <=3} be the vector space of all polynomials of degree 3 or less.

a) Explain why the derivative d/dx: V -> V is a linear function

b) Give the matrix for d2/dx2 in the basis {1,x,x^2, x^3} for V

c) Give the matrix for the third derivative d2/dx2: V->V using the same basis

d) Give the matrix for the third derivative d3/dx3: V->V using the same basis

e) Give a basis for ker(d2/dx2)

f) what is the matrix for the linear map (d/dx + 4(d3/dx)): V->V

g) Let T: R[x] ->R[x] be the function T(p(x))= integral from -2x to 2 of p(t)dt

Explain why T is an element of L(R[x])