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

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\section*{Homework week 3}
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Due by the beginning of class on Friday 11th October (hand in via Gradescope).
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\begin{enumerate}[label={\bf\arabic*.}, itemsep=0.8\baselineskip, leftmargin=2em, labelsep=2em, listparindent=0em, parsep=0.3\baselineskip]
  \item Suppose $V, W$ are a finite dimensional inner product spaces and $\phi \colon V \to W$ is a linear map.  Show that if $\phi$ is injective then $\phi^\ast \circ \phi$ is injective (and hence an isomorphism).

  \item Prove that if $V$ is a finite-dimensional inner product space, $\phi \colon V \to V$ is a normal operator and $\lambda, \mu \in F$ are scalars with $\lambda \ne \mu$ then $E(\lambda,\phi) \perp E(\mu, \phi)$ (i.e.\ these subspaces are orthogonal).

  \item Suppose $V$ is a finite-dimensional vector space, $\phi \colon V \to V$ is a diagonalizable linear map, and $U \subseteq V$ is a $\phi$-invariant subspace.

    Prove that the restriction $\phi|_U \colon U \to U$ is diagonalizable.

    [{\bf Hint}: the maps $\Phi_i$ from the proof of Proposition 3.1.7 might come in handy here.]

  \item Suppose $V$ is a finite-dimensional vector space and $\phi, \psi \colon V \to V$ are two linear maps such that $\phi \circ \psi = \psi \circ \phi$ (i.e., the maps commute).
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item For any $\lambda \in F$, prove that $E(\lambda, \phi)$ is $\psi$-invariant.
      \item Suppose that $\phi$, $\psi$ are diagonalizable.  Prove that there is a basis $B$ for $V$ consisting of simultaneous eigenvectors of $\phi$, $\psi$.

        [{\bf Hint}: remember Q3.]
    \end{enumerate}

  \item Let $V$ be a finite-dimensional inner product space 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 Show that $\lambda$ is an eigenvalue of $V$ if and only if $\overline{\lambda}$ is an eigenvalue of $\phi^\ast$.
      \item Show that $\phi$ is diagonalizable if and only if $\phi^\ast$ is diagonalizable.
    \end{enumerate}

  \item Let $V$, $W$ be finite-dimensional inner product spaces, and $\phi \colon V \to W$ 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, for any $\lambda \in F$, $\lambda \ne 0$, $\dim E(\lambda, \phi^\ast \circ \phi) = \dim E(\lambda, \phi \circ \phi^\ast)$.
      \item Determine the relationship between $\dim E(0, \phi^\ast \circ \phi)$ and $\dim E(0, \phi \circ \phi^\ast)$.
     \end{enumerate}

  \item My office has three chairs, arranged in a row.\footnote{As some of you know, my office doesn't actually have chairs in it right now.  Really, sorry about that.}  Every hour, I roll a $4$-sided die and move chairs, or not, according to the following rule:
    \begin{itemize}
      \item if I'm in an end chair, I stay put if I roll a 1, 2 or 3 and move to the middle if I roll a 4;
      \item if I'm the middle chair, I move left on a 1 roll, stay put on a 2 or 3 roll and move right on a 4 roll.
    \end{itemize}
    So, if my probabilities of being in each chair one hour are $(p_L, p_M, p_R)$, then the probabilities for the next hour are given by
    \[
      \phi(p_L, p_M, p_R) = \left(\frac3{4} p_L + \frac1{4} p_M, \frac1{4} p_L + \frac1{2} p_M + \frac1{4} p_R, \frac1{4} p_M + \frac3{4} p_R \right) \ .
    \]

  I start in the left chair.  Show that the probability I am in the left chair again after 1000 straight hours in my office is $1/3 + \eps$ where $|\eps| \le 2 \cdot (3/4)^{1000}$ (which is around $2.3 \cdot 10^{-125}$).
\end{enumerate}

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