91Ó°ÊÓ

Define \(T \in \mathcal{L}\left(\mathbf{C}^{2}\right)\) by \\[T(w, z)=(z, 0).\\] Find all generalized eigenvectors of \(T\).

Define \(T \in \mathcal{L}\left(\mathbf{C}^{2}\right)\) by \\[T(w, z)=(-z, w).\\] Find all generalized eigenvectors of \(T\).

Suppose \(T \in \mathcal{L}(V), m\) is a positive integer, and \(v \in V\) is such that \(T^{m-1} v \neq 0\) but \(T^{m} v=0 .\) Prove that \\[\left(v, T v, T^{2} v, \ldots, T^{m-1} v\right)\\] is linearly independent.

Suppose \(T \in \mathcal{L}\left(\mathbf{C}^{3}\right)\) is defined by \(T\left(z_{1}, z_{2}, z_{3}\right)=\left(z_{2}, z_{3}, 0\right) .\) Prove that \(T\) has no square root. More precisely, prove that there does not exist \(S \in \mathcal{L}\left(\mathrm{C}^{3}\right)\) such that \(S^{2}=T\).

Suppose \(S, T \in \mathcal{L}(V) .\) Prove that if \(S T\) is nilpotent, then \(T S\) is nilpotent.

Suppose \(V\) is an inner-product space. Prove that if \(N \in \mathcal{L}(V)\) is self-adjoint and nilpotent, then \(N=0\).

Suppose \(N \in \mathcal{L}(V)\) is such that null \(N^{\text {dim } V-1} \neq\) null \(N^{\text {dim } V}\). Prove that \(N\) is nilpotent and that \\[\operatorname{dim} \text { null } N^{j}=j\\] for every integer \(j\) with \(0 \leq j \leq \operatorname{dim} V\).

Suppose \(T \in \mathcal{L}(V)\) and \(m\) is a nonnegative integer such that range \(T^{m}=\) range \(T^{m+1}\). Prove that range \(T^{k}=\) range \(T^{m}\) for all \(k > m\).

Prove or give a counterexample: if \(T \in \mathcal{L}(V),\) then \(V=\) null \(T \oplus\) range \(T\).

Suppose \(V\) is a complex vector space, \(N \in \mathcal{L}(V),\) and 0 is the only eigenvalue of \(N .\) Prove that \(N\) is nilpotent. Give an example to show that this is not necessarily true on a real vector space.