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\lhead{{\sf Summer Bridge Course (Algebra)}: Monday July 20}
\chead{}\rhead{\thepage}
\lfoot{Due Tuesday, July 21 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
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\begin{document}

%\(\;\) \vspace{-4mm}


\noindent \textbf{Topics}: The dual of a vector space. \\

\noindent \textbf{Do}: Answer the following questions. \\

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

% HW 10, Problem 1
\begin{exercise}
  Let \(X\) be a vector space over a field \(K\) and let \(X'\) be the set of linear functions from \(X\) to \(K\), also known as the \emph{dual space} of \(X\). 
  \begin{enumerate}[label=(\roman*)]
  \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). \\
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}

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

% HW 10, Problem 2
\begin{exercise}
  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}[label=(\roman*)]
  \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
    \(\mathcal{P}'_2\).
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}

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