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\title{MATH 202A \\ Applied Algebra I \\ Fall 2019}

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\section*{Homework week 7}
\vspace{0.5\baselineskip}

Due by the beginning of class on Friday 15th 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 Suppose $V$ is a finite-dimensional vector space, $\phi \colon V \to V$ is a nilpotent linear map, $m \ge 1$, $n_1,\dots,n_m \ge 1$ are integers ($n_1+\dots+n_m = \dim V$) and $v_{i,j}$ ($1 \le i\le m$, $1 \le j \le n_i$) are some basis 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 Theorem 4.2.7.

    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Prove that
        \[
          \dim \ker (\phi^k) = \sum_{i=1}^m \min(k, n_i).
        \]
      \item Hence, for each integer $r \ge 1$, find an expression for $|\{ i \colon 1 \le i \le m,\ n_i = r\}|$ (the number of blocks of size $r$) in terms of the quantities $\dim \ker (\phi^k)$.
    \end{enumerate}

    [In other words, this shows nilpotent normal form is unique up to permuting the blocks.]

  \item Consider the space $V$ of complex sequences $(a_0,a_1,a_2,\dots)$ satisfying
    \[
      a_n = r_1 a_{n-1} + r_2 a_{n-2} + \dots + r_d a_{n-d}
    \]
    for all $n \ge d$, where $d \ge 1$ is an integer and $r_1,\dots,r_d \in \CC$ are fixed constants.

    Show that there is a basis for $V$ consisting of sequences of the form
        \[
          a_n = \lambda^n \binom{n}{\ell}
        \]
      for some $\lambda \in \CC$ and $\ell \ge 0$. 

      [{\bf Hint}: let $\phi \colon V \to V$ denote the (infinite) left shift, as in Pset 6 Q6.  One approach is to consider $v=(a_0,a_1,\dots) \in G(\lambda, \phi)$ and find an explicit formula for $a_n$.]

  \item 
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Suppose $V$ is an inner product space with inner product $\langle -, - \rangle$ and associated norm $\|\cdot\|$.
        Prove that the \emph{parallelogram law} holds for $\|\cdot\|$: that is, for all $v,w \in V$ we have
        \[
          \|v + w\|^2 + \|v - w\|^2 = 2 \|v\|^2 + 2 \|w\|^2.
        \]
      \item For any $n \ge 1$ and $1 \le p \le \infty$, $p \ne 2$, find vectors $v,w \in \RR^n$ such that
        \[
          \|v + w\|_p^2 + \|v - w\|_p^2 \ne 2 \|v\|_p^2 + 2 \|w\|_p^2.
        \]
    \end{enumerate}

  \item 
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Suppose $V$ is a finite-dimensional real vector space and $\|\cdot\|$ is a norm on $V$ obeying the parallelogram law: that is,
        \[
          \|v + w\|^2 + \|v - w\|^2 = 2 \|v\|^2 + 2 \|w\|^2
        \]
        for all $v,w \in V$.  Prove that there is an inner product $\langle -,- \rangle$ on $V$ such that $\|v\| = \sqrt{\langle v,v \rangle}$.
        
        [{\bf Hint}: consider the quantity $\frac14 \left( \|v+w\|^2 - \|v-w\|^2 \right)$.]

      \item Prove the same statement as (i) assuming now that $V$ is a complex vector space.

        [{\bf Hint}: it may now help to build an expression from the quantities $\|v \pm w\|^2$ and $\|v \pm i w \|^2$.]
    \end{enumerate}

  \item Let $V = \RR^n$, and identify $V^\ast = \RR^n$ in the usual way; i.e., a vector $(a_1,\dots,a_n) \in \RR^n$ corresponds to a linear map $\RR^n \to \RR$,
    \[
      (x_1,\dots,x_n) \mapsto a_1 x_1 + \dots + a_n x_n.
    \]
    For each of the norms $\|\cdot\|_\infty$ and $\|\cdot\|_1$ on $\RR^n$, determine carefully the value of the corresponding dual norms
      $\|(a_1,\dots,a_n)\|_\infty^\ast$, $\|(a_1,\dots,a_n)\|_1^\ast$ for all $(a_1,\dots,a_n) \in \RR^n \cong V^\ast$.
\end{enumerate}
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