91Ó°ÊÓ

Suppose \(T \in \mathcal{L}(V) .\) Prove that if \(U_{1}, \ldots, U_{m}\) are subspaces of \(V\) invariant under \(T,\) then \(U_{1}+\cdots+U_{m}\) is invariant under \(T\).

Suppose \(T \in \mathcal{L}(V)\). Prove that the intersection of any collection of subspaces of \(V\) invariant under \(T\) is invariant under \(T\).

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

Define \(T \in \mathcal{L}\left(\mathbf{F}^{3}\right)\) by \\[ T\left(z_{1}, z_{2}, z_{3}\right)=\left(2 z_{2}, 0,5 z_{3}\right). \\] Find all eigenvalues and eigenvectors of \(T\).

Suppose \(n\) is a positive integer and \(T \in \mathcal{L}\left(\mathbf{F}^{n}\right)\) is defined by \\[ T\left(x_{1}, \ldots, x_{n}\right)=\left(x_{1}+\cdots+x_{n}, \ldots, x_{1}+\cdots+x_{n}\right); \\] in other words, \(T\) is the operator whose matrix (with respect to the standard basis) consists of all 1 's. Find all eigenvalues and eigenvectors of \(T\).

Find all eigenvalues and eigenvectors of the backward shift op. erator \(T \in \mathcal{L}\left(\mathbf{F}^{\infty}\right)\) defined by \\[ T\left(z_{1}, z_{2}, z_{3}, \dots\right)=\left(z_{2}, z_{3}, \dots\right) \\].

Suppose \(T \in \mathcal{L}(V)\) is invertible and \(\lambda \in \mathbf{F} \backslash\\{0\\} .\) Prove that \(\lambda\) is an eigenvalue of \(T\) if and only if \(\frac{1}{x}\) is an eigenvalue of \(T^{-1}\).

Suppose \(S, T \in \mathcal{L}(V) .\) Prove that \(S T\) and \(T S\) have the same eigenvalues.

Suppose \(T \in \mathcal{L}(V)\) is such that every vector in \(V\) is an eigenvector of \(T .\) Prove that \(T\) is a scalar multiple of the identity operator.

Suppose \(T \in \mathcal{L}(V)\) is such that every subspace of \(V\) with di mension \(\operatorname{dim} V-1\) is invariant under \(T .\) Prove that \(T\) is a scalar multiple of the identity operator.