Rigel Kent

Part One. Motivating example Suppose that \(\alpha, \beta\) are the roots to the following quadratic equation $$x^2 + qx + p = 0$$ Following Viete's theorem, we can directly express \(\alpha + \beta\) and \(\alpha \beta\) in terms of the coefficients of the equation, and they can be written as $$ \left\{ \begin{array}{l} \alpha + \beta = -q\\ \alpha \beta = p \end{array}\right. $$ We know that \(\alpha\) and \(\beta\) can be written as linear combination of \(\alpha + \beta\) and \(\alpha - \beta\): $$\begin{pmatrix}\alpha \\ \beta \end{pmatrix} = \mathbf{a} ( \alpha + \beta) + \mathbf{b} (\alpha - \beta) $$ where \(\mathbf{a} = \begin{pmatrix}\frac{1}{2} & \frac{1}{2}\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}\frac{1}{2} & -\frac{1}{2}\end{pmatrix}\). We also know that \((\alpha - \beta)^2\) can be written in terms of \(p\...