1.8.5 Quadratic forms

Problem: For the following forms, find a non-degenerate linear transformation that takes the form \( f(x) \) in the form \( g(y) \) (the desired transformation is defined ambiguously). \[ \begin{array}{l} f(x)=2 x_{1}^{2}+9 x_{2}^{2}+3 x_{3}^{2}+8 x_{1} x_{2}-4 x_{1} x_{3}-10 x_{2} x_{3}, \\ g(y)=2 y_{1}^{2}+3 y_{2}^{2}+6 y_{3}^{2}-4 y_{1} y_{2}-4 y_{1} y_{3}+8 y_{2} y_{3} . \end{array} \]