Question: Consider the linear operator T : P3(R) P3(R), a + bx + cx2 + dx3 7 b + cx + dx2 . (i) (2 points)

Consider the linear operator T : P3(R) P3(R), a + bx + cx2 + dx3 7 b + cx + dx2 . (i) (2 points) If B = 1, x, x2 , x3 is the standard basis, determine MB(T).

(ii) (4 points) If T 2 = T T, find a basis for ker(T 2 ).

(iii) (4 points) Endow P3(R) with the inner product (p, q) = mini 0 to max3 p(i)q(i). Find an orthonormal basis for ker(T 2 ).

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