Determinant of Polynomial

Question

I was reading some paper and it says 'Let $\Delta$ denote the determinant of the polynomials $P,Q$ and $R$ with respect to the basis $1,X,X^2$' ($P,Q$ and $R$ are degree 2 polynomials). And then I suddenly realise I don't actually seem to know what exactly does that mean -- an educated guess would be the determinant of the $3\times 3$ matrix formed by the coefficients of the polynomials but can someone confirm that? Strange nothing obvious comes up in Google...

Thanks!

Answer 1

It probably means just what you said: Write the coordinates of $P$, $Q$ and $R$ in the given basis. (These are just the coefficients, ordered from lowest to highest powers.) You get three elements of $\mathbb{R}^3$. Putting them together, you get a $3 \times 3$ matrix. Take the determinant of that.

Answer 2

If the determinant thus defined is non zero, the polynomials $P,Q, R$ thus defined are linearly independent, so they are a basis for the vector space of polynomials with degree at most 2.