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Let P(R) denote the family of all polynomials (in a single variable x) with real coefficients....

Let P(R) denote the family of all polynomials (in a single variable x) with real coefficients. We have seen that with the operations of pointwise addition and multiplication by scalars, P(R) is a vector space over R. Consider the 2 linear maps D, I : P(R) to P(R), where D is differentiation and I is anti-differentiation. In detail, for a polynomial p = a0+a1x1+...+anxn, we have D(p) = a1+2a2x+....+nanxn-1 and I(p) = a0x+(a1/2)x2+...+(an/(n+1))xn+1.

a. Show that D composed with I = the identity of P(R).

b. Show that neither D nor I is an isomorphism.

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