Question: Consider the vector spaces Pn[a, b] of all polynomials with degree at most n and C[a, b] of all continuous functions, both only on

Consider the vector spaces Pn[a, b] of all polynomials with degree at most n and C[a, b] of all continuous functions, both only on the interval [a, b], where a < 0 < b. Let : Pn[a, b] C[a, b] be the transformation that maps a polynomial p(t) to an antiderivative P(t) satisfying P(0) = 0. That is, fp(t) = P(t), and P'(t) = p(t). (a) Show that is a linear transformation. (b) Determine the kernel of f. (c) Determine the range of f (think carefully).

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