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\fancyhead[L]{\textbf{MATH 510 Homework 4}}
\fancyhead[R]{\textbf{Your-Name-Goes-Here}}
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{\small
\begin{itemize}
\item Homework 4 is due February 16 at the beginning of class.
\item Write the problem statement followed by a proof or solution.
\item List problems in the same order they were given.
\item If you skip a problem, include the problem statement with no solution. 
\end{itemize}}

\begin{enumerate}[{\rm 1.}]
\item Let $T:\mathbb{R}^2\to\mathbb{R}^2$ be the linear operator whose matrix in the standard ordered basis of $\mathbb{R}^2$ is
\[A=\begin{bmatrix}
 1 & -1 \\ 2 & 2
 \end{bmatrix}.
\]
\begin{enumerate}[{\rm (a)}]
\item Prove that the only subspaces of $\mathbb{R}^2$ invariant under $T$ are $\mathbb{R}^2$ and the zero subspace.
\item Let $U:\mathbb{C}^2\to\mathbb{C}^2$ be the linear map whose matrix in the standard ordered basis of $\mathbb{C}^2$ is $A$. Show that $U$ does have $1$-dimensional invariant subspaces.
\end{enumerate}
\smallskip
\item Let $V$ be a finite-dimensional vector space and $W$ be an invariant subspace for a linear map $T:V\to V$. Prove that the minimal polynomial for the restricted linear map $T\big|_W:W\to W$ divides the minimal polynomial for $T$.
\smallskip
\item Let $V$ be the space of $n\times n$-matrices over a field $\mathbb{F}$, and let $A$ be a fixed $n\times n$-matrix over $\mathbb{F}$. Define a linear operator $T$ on $V$ by $T(B) = AB - BA$. Prove that if $A$ is a nilpotent matrix, then $T$ is a nilpotent operator.
\smallskip
\item Let $V$ be a finite-dimensional vector space and let $T,S\in\operatorname{End}(V)$ be commuting linear maps. Suppose there are ordered bases $\mathcal{B}$ and $\mathcal{C}$ for $V$ such that $[T]_{\mathcal{B}}$ and $[S]_{\mathcal{C}}$ are diagonal. Show that there is an ordered basis $\mathcal{D}$ for $V$ such that $[T]_{\mathcal{D}}$ and  $[S]_{\mathcal{D}}$ are diagonal.
\smallskip
\item Let $A$ be an $n\times n$-matrix with real entries such that $A^2+I=0$. Prove that $n=2k$ for some $k>0$, and that furthermore $A$ is similar to a matrix of the block form
\[\begin{bmatrix}0_k & -I_k\\ I_k& 0_k\end{bmatrix}.\]
\end{enumerate}
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