Isomorphism mapping from $\mathbb R^3$ to $\mathbb P_2(\mathbb R)$

Question

Can you give an example of an isomorphism mapping from $\mathbb R^3 \to \mathbb P_2(\mathbb R)$(degree-2 polynomials)?

I understand that to show isomorphism you can show both injectivity and surjectivity, or you could also just show that an inverse matrix exists.

My issue is that I don't think you can represent the transformation with a matrix because of the polynomial space.

How would you come to proving isomorphism without the use of matrix representations?

Answer 1

What about$$\begin{array}{rccc}\psi\colon&\mathbb R^3&\longrightarrow&P_2(\mathbb R)\\&(a,b,c)&\mapsto&a+bx+cx^2?\end{array}$$It is linear, injective and surjective.

Answer 2

Assuming you mean the polynomials of degree less than or equal to $2$, it is a three dimensional space, with basis $\{1,x,x^2\}$. So, just send basis vectors to basis vectors:

$$e_1\to1,e_2\to x,e_3\to x^2$$.

The matrix, rel these two standard bases, is the identity.