Let pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. let d2: p3→p1 be the function that sends a polynomial to its second derivative. that is, d2(p(x))=p′′(x) for all polynomials p(x)∈p3. is d2 a linear transformation?

Yes. A trasformation is linear if:

The (second) derivative satisfies both these request: in fact, a generic element in p3 is . Let's prove that, for any real number , .

We have:

On the other hand,

The proof of the second point is the same: write the derivative of the sum, and the sum of the derivaties. You'll see that they are the same polynomial.

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step-by-step explanation: