## The Coordinate Vector for a Polynomial with respect to the Given Basis

## Problem 683

Let $\mathrm{P}_3$ denote the set of polynomials of degree $3$ or less with real coefficients. Consider the ordered basis

\[B = \left\{ 1+x , 1+x^2 , x – x^2 + 2x^3 , 1 – x – x^2 \right\}.\]
Write the coordinate vector for the polynomial $f(x) = -3 + 2x^3$ in terms of the basis $B$.