Diagonalize a symmetric matrix associated to a form

Solution to Linear Algebra Hoffman & Kunze Chapter 9.2 Exercise 9.2.5
Solution: The matrix of $f$ in the standard basis is given by $\begin{pmatrix}1 & 2\\2& 4\end{pmatrix}$. Its  eigenvalues are zero and 5. The corresponding eigenvectors are $(2,-1)$ and $(1,2)$. Now it is easy to check that the matrix of $f$ in the basis $(2,-1)$ and $(1,2)$ is diagonal $\mathrm{diag}(0,25)$.