We show that isomorphism's can be tailored to fit in that, sometimes, given vectors in the domain

Question:

We show that isomorphism's can be tailored to fit in that, sometimes, given vectors in the domain and in the range we can produce an isomorphism associating those vectors.
(a) Let B = (1, 2, 3,) be a basis for P2 so that any ˆˆ P2 has a unique representation as = c11 + c22 + c33, which we denote in this way.

Show that the RepB(ˆ™) operation is a function from P2 to R3 (this entails showing that with every domain vector 2 P2 there is an associated image vector in R3, and further, that with every domain vector 2 P2 there is at most one associated image vector).
(b) Show that this RepB(ˆ™) function is one-to-one and onto.
(c) Show that it preserves structure.
(d) Produce an isomorphism from P2 to R3 that fits these specifications.