Question:
Linear transformation in Matrix form
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.