25 Suppose that $T \in \mathcal{L}(V)$ and there is an orthonormal basis $e_1, \dots, e_n$ of $V$ consisting of eigenvectors of $T$, with corresponding eigenvalues $\lambda_1, \dots, \lambda_n$. Show that if $k \in \{1, \dots, n\}$, then the pseudoinverse $T^\dagger$ satisfies the equation
$$
T^\dagger e_k = \begin{cases}
\frac{1}{\lambda_k} e_k & \text{if } \lambda_k \ne 0, \\
0 & \text{if } \lambda_k = 0.
\end{cases}
$$
