
Quadractic forms
suppose i'm asked to find the associated bilinear form for the quadratic form $\displaystyle q(v)=x_1x_3+x_2^2,$ wonder if i first need to find the associated matrix to this quadratic form, then in order to get the associated bilinear form i'd do $\displaystyle f\big((x_1,x_2,x_3),(y_1,y_2,y_3)\big)=(x_1~x_2~x_ 3)[\text{matrix of the quadratic form}](y_1~y_2~y_3)^t$ ?

There are a number of different ways to write a matrix representing a quadratic form the standard way is to use a symmetric matrix.
You want $\displaystyle \begin{pmatrix}x_1 & x_2 & x_3\end{pmatrix}\begin{pmatrix}a & b & c \\ b & d & e \\ c & e & f\end{pmatrix}\begin{pmatrix}x_1 \\ x_2 \\ x_3\end{pmatrix}= x_1x_3+ x_2^2$ for all $\displaystyle x_1$, $\displaystyle x_2$, and $\displaystyle x_3$.
Multiply the left side and set it equal to the right. it should be easy to determine a, b, c, d, e, and f (most of them will be 0).