. The following polynomials are the first five in the sequence known as Chebyshev polynomials of the first kind

T0(x)=1, T1(x)=x, T2(x)=2x2 −1 T3(x)=4x3 −3x, T4(x)=8x4 −8x2 +1.

(a) Show that {T0, T1, T2, T3, T4} is a basis for P4, the space of polynomials of degree ≤ 4.
(b) Check that differentiation defines a linear transformation TD : P4 → P3 and write down the matrix of each linear transformation in the Chebyshev basis. Similarly, check that integration is a linear transformation TS : P3 → P4.
(c) Let D and S be the differentiation and integration matrices, respectively, from part (b). Compute the matrix products DS and SD. Interpret the results using calculus: choose a suitable polynomial in P4, differentiate it, and then integrate it.
(d) Write down bases for the null spaces and column spaces of D and S. Provide the cor- responding polynomials. Can you interpret your results about D and S in light of what you know about differentiation and integration from calculus?