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Linear transformation in Matrix form

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Let P3 denote the real vector space of polynomial functions of degree up to 3, i.e.

p3-{f(x)=a3x^3+a2x^2+a1x| aiER}

Consider the linear transformation D: P3--> P3 given by the derivative D(f) = d/dx f
a) What is the kernel of D? Give a basis of the kernel
b) The set B={1, x, x^2, x^3} forms a basis of D. Write the matrix Mb^b (D)
c) What is the rank of D
d) Show that the 4-fold composition D^4 = D*D*D*D (i.e. applying D to the vector space four times) is the zero map.

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The solution shows in detail, how to find the matrix representation of the derivative transformation in the basis of polynomials of degree 3, how to find its kernel and rank and showing that applied 4 times it indeed becomes the zero map.

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