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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due September 3
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2021 $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-3mm}

\noindent\emph{Read}: Lax, Chapter 2, pages 13--18. \medskip

\begin{enumerate}

%%------------------------------------------------------------------------
  
\item Let $K$ be a finite field. The \emph{characteristic} of $K$,
  denoted $\Char K$, is the smallest positive integer $n$ for which
  $n1:=\underbrace{1+1+\cdots+1}_{\tiny\mbox{$n$ times}}=0$.
  \begin{enumerate}
  \item Prove that the characteristic of $K$ is prime. 
  \item Show that $K$ is a vector space over $\Z_p$, where $p=\Char K$.
  \item Show that $|K|$ is a prime power.
  \item Show that if $K$ and $L$ are finite fields with $K\subset
    L$ and $|K|=p^m$ and $|L|=p^n$, then $m$ divides $n$. \medskip
  \end{enumerate}

%%------------------------------------------------------------------------

\item Let $X$ be a vector space over a field $K$ and let $X'$ be the
  the set of linear functions from $X$ to $K$, also known as the
  \emph{dual space} of $X$. 
  \begin{enumerate}
  \item Let $x_1,\dots,x_n$ be a basis for $X$. Define $\ell_j\in X'$
    by $\ell_j(x_i)=\delta_{ij}$. Show that $\ell_1,\dots,\ell_n$ is a
    basis for $X'$; it is called the \emph{dual basis} of
    $x_1,\dots,x_n$.
  \item Find the dual basis of $x_1=(1,-1,3)$, $x_2=(0,1,-1)$, and
    $x_3=(0,3,-2)$ in $X=\R^3$.
  \item Express the scalar function $f\in X'$, where
    $f(x,y,z)=2x-y+3z$ as a linear combination of the dual basis,
    $\ell_1,\ell_2,\ell_3$, from Part~(b). \medskip
  \end{enumerate}

%%------------------------------------------------------------------------

\item Let $S$ be a subset of $X$. The \emph{annihilator} of $S$ is the
  set 
  \[
  S^\perp=\{\ell\in X'\mid\ell(s)=0\;\;\mbox{for all }s\in S\}\,.
  \]
  \begin{enumerate}
  \item Show that $\Span(S)$ is the intersection of all subspaces
    $T_\alpha$ of $X$ that contain $S$:
    \[
    \displaystyle\Span(S)=\bigcap_{S\subseteq T_\alpha\leq X} T_\alpha, 
    \]
    making it well-founded to speak of the ``\emph{smallest subpace of $X$ that
      contains $S$}.''
  \item Show that $S^\perp$ is a subspace of $X'$, and that
    $S^\perp=\Span(S)^\perp$. \medskip
  \end{enumerate}

%%------------------------------------------------------------------------

\item Let $\mathcal{P}_2$ be the vector space of all polynomials
  $p(x)=a_0+a_1x+a_2x^2$ over $\R$, with degree $\leq 2$. Let
  $\xi_1,\xi_2,\xi_3$ be distinct real numbers, and define
  \[
  \ell_j\colon\mathcal{P}_2\longrightarrow\R,\qquad
  \ell_j(p)=p(\xi_j)\quad\mbox{for}\quad j=1,2,3\,.
  \]
  \begin{enumerate}
  \item Show that $\ell_1,\ell_2,\ell_3$ is a basis for the dual space
    $\mathcal{P}'_2$.
  \item Find polynomials $p_1(x),p_2(x),p_3(x)$ in $\mathcal{P}_2$ of
    which $\ell_1,\ell_2,\ell_3$ is the dual basis in
    $\mathcal{P}'_2$. \medskip
  \end{enumerate}

%%------------------------------------------------------------------------

\item Let $W$ be the subspace of $\R^4$ spanned by $(1,0,-1,2)$ and
  $(2,3,1,1)$. Find the annihilator of $W$. Write your answer by
  giving an explicit basis of $W^\perp$ in terms of the dual basis
  vectors $\ell_1,\dots,\ell_4$.

%%------------------------------------------------------------------------

\end{enumerate}

\end{document}
