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\title{MATH 202A \\ Applied Algebra I \\ Fall 2021}

\begin{document}

\maketitle

\def\id{\operatorname{id}}
\def\im{\operatorname{im}}
\def\ker{\operatorname{ker}}
\def\eps{\varepsilon}
\def\spec{\operatorname{spec}}

\vspace{-1.5\baselineskip}
\section*{Homework week 6}
\vspace{0.5\baselineskip}

Due by 2359 on Sunday 7th November (hand in via Gradescope).
\vspace{0.5\baselineskip}

\begin{enumerate}[label={\bf\arabic*.}, itemsep=0.8\baselineskip, leftmargin=2em, labelsep=2em, listparindent=0em, parsep=0.3\baselineskip]
  \item Let $V$ be a finite-dimensional vector space over a field $F$ and $\phi \colon V \to V$ a linear map.
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Prove that if $\phi$ has an eigenvalue $\lambda \ne 0$ then $\phi$ is not nilpotent.
      \item Prove that, if $F = \CC$ and $\spec(\phi) = \{0\}$ then $\phi$ is nilpotent.  [For this question, do not use Jordan Normal Form itself, though you may use facts that go into it.]
      \item If $F = \RR$ and $V = \RR^3$, find an example of a map $\phi$ such that $\spec(\phi) = \{0\}$ but $\phi$ is not nilpotent.
    \end{enumerate}

  \item Let $V$ be a finite-dimensional vector space and $\phi \colon V \to V$ a linear map.

    Consider the subspaces
    \[
      \{0\} \subseteq \ker \phi \subseteq \ker (\phi^2) \subseteq \dots
    \]
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Prove that if $\ker (\phi^k) = \ker (\phi^{k+1})$, then in fact $\ker \phi^{\ell} = \ker \phi^{\ell+1}$ for all $\ell \ge k$.
      \item Suppose $\phi$ is nilpotent.  Prove that $\phi^r = 0$ holds for some $r \le \dim V$.  [Do not use the structure theorem of nilpotent maps to answer this part.]
    \end{enumerate}

  \item Let $V$ be a finite dimensional vector space and $\phi \colon V \to V$ a linear map.  Suppose $v \in V$, $v \ne 0$, and suppose that for some non-negative integer $m \ge 0$ we have
    \[
      \phi^m(v) \ne 0
    \]
    but
    \[
      \phi^{m+1}(v) = 0 \ .
    \]
    Show that $v,\phi(v),\dots,\phi^m(v)$ are linearly independent.

  \item Suppose $V$ is a finite-dimensional vector space and $\phi \colon V \to V$ is nilpotent.
    Suppose $n = \dim V$ and $n_1,\dots,n_m \ge1$ are integers such that $n_1+\cdots+n_m=n$ and there is a basis $B=(v_{i,j})_{i\in[m], j \in [n_i]}$ for $V$ such that
    \[
      \phi(v_{i,j}) = \begin{cases} v_{i,j-1} &\colon j > 1 \\ 0 &\colon j=1 \end{cases}
    \]
    as guaranteed by the structure theorem for nilpotent maps.
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item For $k \ge 1$, prove that
        \[
          \dim \ker (\phi^k) = \sum_{i=1}^m \min(k, n_i).
        \]
      \item Hence, or otherwise, show that for $\ell \ge 1$:
        \[
          \bigl\lvert \bigl\{ i \colon 1 \le i \le m,\ n_i = \ell \bigr\} \bigr\rvert = 2 \dim \ker(\phi^{\ell}) - \dim \ker(\phi^{\ell+1}) - \dim \ker(\phi^{\ell-1})
        \]
        where by convention $\phi^0 = \id_V$.
    \end{enumerate}
    [This establishes uniqueness in the structure theorem for nilpotent maps: the number of blocks of each size is determined by values that clearly only depend on $\phi$.
    If you like, use this to deduce uniqueness for the number and sizes of blocks in Jordan Normal Form.]

  \item 
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Let $V$ be finite dimensional and let $\psi \colon V \to V$ be a nilpotent map, where $\psi^k = 0$.  Write $\phi = \psi + \lambda \id$ for some $\lambda \in F$. Show that for any integer $m \ge 0$,
        \[
          \phi^m = \sum_{r=0}^{\min(k-1,m)} \lambda^{m-r} \binom{m}{r} \psi^r.
        \]
      \item For any integer $m \ge 0$, compute the matrix $J_k^m$.  Using (a), or otherwise, also compute $J(k,\lambda)^m$.  Here
        \[
          J_k = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 & 0 \\
          0 & 0 & 1 & \cdots & 0 & 0 \\
          \vdots \\
          0 & 0 & 0 & \cdots & 0 & 1 \\
          0 & 0 & 0 & \cdots & 0 & 0
          \end{pmatrix}, \ \ \ 
          J(k,\lambda) = \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0 & 0 \\
          0 & \lambda & 1 & \cdots & 0 & 0 \\
          \vdots \\
          0 & 0 & 0 & \cdots & \lambda & 1 \\
          0 & 0 & 0 & \cdots & 0 & \lambda
        \end{pmatrix}
        \]
        are $k \times k$ matrices.
    \end{enumerate}

  \item Consider the vector space of infinite complex sequences $(a_0,a_1,a_2,\dots)$.  Let $V$ denote the subspace that satisfy the homogeneous recursion relation
    \[
      a_n = 6 a_{n-1} - 12 a_{n-2} + 8 a_{n-3}
    \]
    for all $n \ge 3$.
    [You need not verify that this is a subspace.]  Also let $\phi \colon V \to V$ denote the ``infinite left shift'' map
    \[
      \phi(a_0,a_1,a_2,\dots) = (a_1,a_2,a_3,\dots).
    \]
    You should satisfy yourself that this latter sequence is indeed in $V$, but do not need to write anything.

    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Prove that the linear map $\psi \colon V \to \CC^3$, $\psi(a_0,a_1,a_2,\dots) = (a_0,a_1,a_2)$ is an isomorphism, and write down a matrix for $\phi$ with respect to the basis $B=\psi^{-1}(e_1),\psi^{-1}(e_2),\psi^{-1}(e_3)$ where $e_1,e_2,e_3$ is the standard basis for $\CC^3$.
      \item Hence, or otherwise, compute $\spec(\phi)$, and deduce that $\phi - 2 \id_V$ is nilpotent.
      \item Using Q4(i), or otherwise, and noting that $a_n$ is the first entry of $\phi^n(a_0,a_1,\dots)$, find a closed-form formula for $a_n$ in terms of $a_0,a_1,a_2$.
    \end{enumerate}

\end{enumerate}
\end{document}
