1. We need the following definition for part (c) below. Vector spaces V and W are isomorphic if there is a linear transformation o: V → W that is a bijection, in the sense that (i) (u) = $(v) = u = v for all u, v€ V; and (ii) for all w e W there exists v € V such that $(v) = w. Consider the set P2 := {f: R →R: f (x) = a.x² + ba + c for some a, b, c E R}. So P2 is the set of polynomials of degree up to 2. This set is a vector space over R under addition given by %3D (a122 + bịa + cı) + (a2a² + b2x + c2) = (a1 + a2)a² + (bi + b2)x + (cı + c2), and scalar multiplication given by X(ax² + bx + c) = \ax² + \bx + Ac. Consider the function D: P2 P2 given by D(ax2² + bx + c) = 2ax + b. (a) Why did I call this function D? (b) Prove that D is a linear transformation'. (c) Find an n such that P2 is isomorphic to R". (Don't just state the value of n; prove why the vector spaces are isomorphic.) (d) Let o : P2 → R" be the function from (c) you used to show that P2 and R" are isomorphic (where n is your value from (c)). Find a matrix Ap such that oo Doo- is given by left multiplication by Ap. (e) Find the eigenvalues and eigenspaces of Ap.