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\title{Math 2135: Linear Algebra}
\author{Assignment 7}
\date{due 5 April 2022, end of day}

\begin{document}
\maketitle

Homework should be submitted as a single PDF attachment to \texttt{theojf@dal.ca}. Please be sure to include your surname in the file name.

You are encouraged to work with your classmates, but your writing should be your own. If you do work with other people, please acknowledge (by name) whom you worked with. You should attempt every question, but it is not expected that you will solve all of them.

\begin{enumerate}
  \item \begin{enumerate}
    \item Define $T \in \cL(\bC^2)$ by $T(w,z) = (z,0)$. Find the generalized eigenspaces of $T$.
    \item Define $T \in \cL(\bC^2)$ by $T(w,z) = (-z,w)$. Find the generalized eigenspaces of $T$.
  \end{enumerate}
  \item Suppose that $T \in \cL(V)$, $m$ is a positive integer, and $v\in V$ is such that $T^{m-1}v \neq 0$ but $T^m v = 0$. Prove that 
  $$ v, Tv, T^2v, \dots, T^{m-1}v$$
  is linearly independent.
  \item Let $T \in \cL(\bC^3)$ be the operator defined by $T(z_1,z_2,z_3) = (z_2,z_3,0)$. Prove that $T$ does not have a square root. In other words, prove that there is no operator $S \in \cL(\bC^3)$ such that $S^2 = T$.
  \item Suppose that $T \in \cL(\bC^4)$ is such that the eigenvalues of $T$ are $3$, $5$, and $8$. Prove that $(T-3)^2 (T-5)^2 (T-8)^2 = 0$.
  \item Suppose that $V$ is a finite-dimensional vector space over $\bF=\bR$. Show that there exists an operator $T \in \cL(V)$ such that $T^2 = -I$ if and only if $\dim V$ is even.
  \item \textbf{Hint for this question:} Read Theorem 8.31 on page 258 of \emph{Linear Algebra Done Right}.
  
  Let $V$ be a finite-dimensional vector space and suppose that $T \in \cL(V)$ is nilpotent. Consider the following formulas:
  $$ \exp(T) := \sum_{k=0}^\infty \frac{1}{k!}T^k, \qquad \log(1 + T) = \sum_{k=1}^\infty \frac{(-1)^{k-1}}k T^k.$$
  \begin{enumerate}
    \item Explain why, even though the formulas look like infinite sums, actually these sums are finite.
    \item Show that $\log(1+T)$ is nilpotent. Show that $\exp(T) - 1$ is nilpotent. \textbf{Hint:} Show that if $T$ is nilpotent and $ST = TS$, then $ST$ is nilpotent.
    \item Show $\exp(\log(1+T)) = 1+T$ and that $\log(\exp(T)) = T$. \textbf{Hint:} Use the fact that the Taylor series of $e^x$ is $\sum_{k=0}^\infty \frac{1}{k!}x^k$ and that the Taylor series of $\ln(1+x)$ is $\sum_{k=1}^\infty \frac{(-1)^{k-1}}k x^k$.
  \end{enumerate}
\end{enumerate}




\end{document}  