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

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\section*{Homework week 2}
\vspace{0.5\baselineskip}

Due by 2359 on Sunday October 10 (hand in via Gradescope).
\vspace{0.5\baselineskip}

\begin{enumerate}[label={\bf\arabic*.}, itemsep=0.8\baselineskip, leftmargin=2em, labelsep=2em,parsep=0.4\baselineskip]
  \item For each of the following, determine whether it is an inner product on $\RR^2$.
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.5\baselineskip]
      \item $\langle (x,y), (x', y') \rangle = x x' - y y'$.
      \item $\langle (x,y), (x', y') \rangle = x x' + x y' + y x' + 2 y y'$.
      \item $\langle (x,y), (x', y') \rangle = x x' + x y' + y x' + y y'$.
    \end{enumerate}

  \item Let $V$ be a finite dimensional inner product space, with inner product $\langle-,-\rangle$ as usual.
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.5\baselineskip]
      \item Suppose $\phi \colon V \to V$ a linear map.  Define a new operation $\langle -,-\rangle_1 \colon V \times V \to F$ by
        \[
          \langle v,w \rangle_1 = \langle \phi(v), \phi(w) \rangle \ .
        \]
        Show that if $\phi$ is invertible then this is an inner product on $V$.
      \item In the set-up of (a), show that if $\phi$ is not invertible then $\langle-,-\rangle_1$ is not an inner product.
      \item Conversely, suppose $\langle -,- \rangle_2$ is yet another inner product on $V$.  Show that there is an invertible linear map $\psi \colon V \to V$ such that
        \[
          \langle v, w \rangle_2 = \langle \psi(v), \psi(w) \rangle \ .
        \]
    \end{enumerate}

  \item Consider the vectors
    \begin{align*}
      v_1 &= (1.1, 1.1, 1.1, 1.1) \\
      v_2 &= (3.3, 3.3, 1.1, 1.1) \\
      v_3 &= (6.6, 4.4, 2.2, 0) \\
      v_4 &= (10, 5.4, 3.2, -1.0) 
    \end{align*}
    in $\RR^4$, which carries the usual dot product.

    You may assume that
    \[
      \begin{pmatrix}
        1.1 & 3.3 & 6.6 & 10 \\ 1.1 & 3.3 & 4.4 & 5.4 \\ 1.1 & 1.1 & 2.2 & 3.2 \\ 1.1 & 1.1 & 0 & -1.0
      \end{pmatrix}
      = 
      \begin{pmatrix}
        0.5 & 0.5 & 0.5 & 0.5 \\
        0.5 & 0.5 & -0.5 & -0.5 \\
        0.5 & -0.5 & 0.5 & -0.5 \\
        0.5 & -0.5 & -0.5 & 0.5
      \end{pmatrix}
      \begin{pmatrix}
        2.2 & 4.4 & 6.6 & 8.8 \\
        0   & 2.2 & 4.4 & 6.6 \\
        0   & 0   & 2.2 & 4.4 \\
        0   & 0   & 0   & 0.2
      \end{pmatrix}.
    \]
    Show that there exist coefficients $a_1, a_2, a_3 \in \RR$ such that
    \[
      \|v_4 - a_1 v_1 - a_2 v_2 - a_3 v_3 \| \le 0.2.
    \]

    [Please tell me if you spot a mistake in these numbers.]

  \item Let $\mathcal P_{\le 4}$ denote the vector space of real-valued polynomials of degree $\le 4$:
    \[
      \mathcal P_{\le 4} = \big\{ p(X) = a_0 + a_1 X + \cdots + a_{4} X^{4} \colon a_0,\dots,a_{4} \in \RR \big\}.
    \]
    You may use without proof that $B = 1,X,X^2,\dots,X^{4}$ and $B'=1, 1+X, (1+X)^2,\dots,(1+X)^{4}$ are two basis for $\mathcal P_{\le 4}$, and that 
    \[
      \phi \colon p(X) \mapsto \frac{d p}{dX}
    \]
    and
    \[
      \psi \colon p \mapsto (X \mapsto p(X+1))
    \]
    are linear maps $\mathcal P_{\le 4} \to \mathcal P_{\le 4}$.  (So e.g.\ $\psi(3+4X) = 3 + 4(X+1) = 7+4X$.)

    Write down:
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.5\baselineskip]
      \item $\mathcal M(\phi, B, B)$;
      \item $\mathcal M(\psi, B, B)$;
      \item $\mathcal M(\id, B', B)$;
      \item $\mathcal M(\id, B, B')$;
      \item $\mathcal M(\phi, B, B')$.
    \end{enumerate}

  \vspace{\baselineskip}
  \noindent
  For a subspace $U \subseteq V$, write
  \[
    U^\perp = \{\phi \in V^\ast \colon \phi(u) = 0 \ \forall u \in U \} \subseteq V^\ast.
  \]
  Similarly, if $W \subseteq V^\ast$, write
  \[
    W^\perp = \{u \in V \colon \phi(u) = 0 \ \forall \phi \in W \} \subseteq V.
  \]

  \item Prove that if $V$ is finite-dimensional (and so we can identify $V$ with $V^{**}$) then $(U^\perp)^\perp = U$.

  \item Let $V$ and $W$ be vector spaces, let $f \colon V \to W$ be a linear map, and let $f^\ast \colon W^\ast \to V^\ast$ be the dual map.

    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.5\baselineskip]
      \item Prove that $\ker (f^\ast) = (\im f)^\perp$.

        \vspace{\baselineskip}
        \noindent Now assume that $V$, $W$ are finite-dimensional (and so we can identify $V$ with $V^{**}$ and $W$ with $W^{**}$).
      \item Prove that $f^{**} = f$.
      \item Prove that $(\im (f^\ast))^\perp = \ker f$.
      \item Prove that $(\ker (f^\ast))^\perp = \im f$.
      \item Prove that $\im (f^\ast) = (\ker f)^\perp$.
    \end{enumerate}

\end{enumerate}

\end{document}
