Exercise 26

We show that isomorphisms 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.

• Let $B=\langle\vec{\beta}_{1},\vec{\beta}_{2},\vec{\beta}_{3}\rangle$ be a basis for $\mathcal{P}_{2}$ so that any $\vec{p}\in\mathcal{P}_{2}$ has a unique representation as $\vec{p}=c_{1}\vec{\beta}_{1}+c_{2}\vec{\beta}_{2}+c_{3}\vec{\beta}_{3}$, which we denote in this way.

  $${\rm Rep}_{B}(\vec{p})=\begin{pmatrix}c_{1}\\ c_{2}\\ c_{3}\end{pmatrix}$$

  Show that the ${\rm Rep}_{B}(\cdot)$ operation is a function from $\mathcal{P}_{2}$ to $\mathbb{R}^{3}$ (this entails showing that with every domain vector $\vec{v}\in\mathcal{P}_{2}$ there is an associated image vector in $\mathbb{R}^{3}$, and further, that with every domain vector $\vec{v}\in\mathcal{P}_{2}$ there is at most one associated image vector).

• Show that this ${\rm Rep}_{B}(\cdot)$ function is one-to-one and onto.

• Show that it preserves structure.

• Produce an isomorphism from $\mathcal{P}_{2}$ to $\mathbb{R}^{3}$ that fits these specifications.

  $$x+x^{2}\mapsto\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}\quad\text{and}\quad 1-x\mapsto\begin{pmatrix}0\\ 1\\ 0\end{pmatrix}$$