Finding representative matrix of quadratic form $ q(x) = x_1x_2 + x_2x_3 + x_1x_3$ with respect to a basis

Question

I'm slightly confused about this question: we have a basis of $R^3$ $B = ${$e_1,e_2, e_3$}, where $x = x_1e_1 + x_2e_2 + x_3e_3$.

We need to give the representative matrix of $q$ with respect to base $B$. I don't understand how this is done; isn't the phrase $x = x_1e_1 + x_2e_2 + x_3e_3$ true for any basis of $R^3$? How can I give the representative matrix with respect to a basis that I do not know? What is it that I am missing?

Any help is appreciated; thanks!

Answer 1

It would be helpful to see the original text of the problem. However, it appears that what you’re being told is that if $(x_1,x_2,x_3)$ are the coordinates of $x$ relative to this (unknown) basis, then $q(x)=x_1x_2+x_2x_3+x_1x_3$. You really don’t need to know the basis elements; it’s enough to know that the $x_i$ in the formula for $q(x)$ are expressed relative to this basis.