Worksheet 21

1. Show that $v$ is eigenvector with eigenvalue $\lambda$ if and only $v \in \mathcal Z(T - \lambda \operatorname{Id}_V)$ and $v \neq 0$.

2. What could possibly be the eigenvectors and eigenvalues for integration? More precisely, for what $f(x)$ is
   \(\int_{-\infty}^y f(x) \ dx = \lambda f(y)\) for some $\lambda \in \mathbb{R}$.

3. Is there a basis of $\mathbb{R}^2$ consisting of eigenvectors for
   \(\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} ?\)