The reason for the expression 1 x − 2 1 2 + x 4 ≠ x − 2 − 1 2 + x − 4 with the proper description.

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Chapter A, Problem 61E

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The reason for the expression 1x−212+x4≠x−2−12+x−4 with the proper description.

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most 2, and let Let P2 denote the vector space of polynomials of degree at D: P2➡ P2 be the transformation that sends a polynomial p(t) = at² + bt+c in P2 to its derivative p'(t) 2at+b, that is, D(p) = p'. (a) Prove that D is a linear transformation. (b) Find a basis for the kernel ker(D) of the linear transformation D and compute its nullity. (c) Find a basis for the image im(D) of the linear transformation D and compute its rank. (d) Verify that the Rank-Nullity Theorem holds for the linear transformation D. (e) Find the matrix representation of D in the standard basis (1,t, t2) of P2.