# A Change of Basis Problem

1. Jul 6, 2008

### e(ho0n3

The problem statement, all variables and given/known data
In $\mathcal{P}_3$ with basis $B = \langle 1 + x, 1 - x, x^2 + x^3, x^2 - x^3 \rangle$ we have this representation.

$$\text{Rep}_B(1 - x + 3x^2 - x^3) = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 2 \end{pmatrix}_B$$

Find a basis $D$ giving this different representation for the same polynomial.

$$\text{Rep}_D(1 - x + 3x^2 - x^3) = \begin{pmatrix} 1 \\ 0 \\ 2 \\ 0 \end{pmatrix}_D$$

The attempt at a solution
I've noticed that

$$1 - x + 3x^2 - x^3 = 1 - x + x^2 + x^3 + 2(x^2 - x^3)$$

so the first and third component of $D$ could be $1 - x + x^2 + x^3$ and $x^2 - x^3$ respectively. I can guess a possible second and fourth component and then check $D$ to determine if it is a basis. Is there an easier way of accomplishing this?