(a) Let $\lambda_{1}, \ldots, \lambda_{d}$ be distinct eigenvalues of an $n \times n$ matrix $A$, with corresponding eigenvectors $\mathbf{v}_{1}, \ldots, \mathbf{v}_{d}$. Prove that the set $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{d}\right\}$ is linearly independent.

(b) Consider the quadric surface $Q$ in $\mathbb{R}^{3}$ defined by

$$2 x^{2}-4 x y+5 y^{2}-z^{2}+6 \sqrt{5} y=0 .$$

Find the position of the origin $\tilde{O}$ and orthonormal coordinate basis vectors $\tilde{\mathbf{e}}_{1}, \tilde{\mathbf{e}}_{2}$ and $\tilde{\mathbf{e}}_{3}$, for a coordinate system $(\tilde{x}, \tilde{y}, \tilde{z})$ in which $Q$ takes the form

$$\alpha \tilde{x}^{2}+\beta \tilde{y}^{2}+\gamma \tilde{z}^{2}=1 .$$

Also determine the values of $\alpha, \beta$ and $\gamma$, and describe the surface geometrically.