For a quadratic form $q(\mathbf{v})$, when you change the basis do you always change the quadratic form? Can you have the same quadratic form with respect to different basis? Or is the quadratic form unique to the basis.
Also, if you're given a quadratic form say $q(\mathbf{v}) = 3x^2 + y^2 - 2z^2 + 4xy - 2xz$, $\:$ and you can clearly deduce the matrix from this $\begin{pmatrix}3&2&-1\\2&1&0\\-1&0&-2\end{pmatrix}$ what is the basis for this quadratic form? Is it unique? Can you deduce it from this? Perhaps with a different basis is there a different 'method' of deducing the matrix?
Thanks!