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\lhead{{\sf Summer Bridge Course (Algebra)}: Wednesday July 22}
\chead{}\rhead{\thepage}
\lfoot{Due Thursday, July 23 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}


\begin{document}

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


\noindent \textbf{Topics}: The matrix of a linear map; change-of-basis
matrices. \\

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

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

% HW 12, Problem 1
\begin{exercise}
Let \(\mathcal{P}_n\) be the vector space of all polynomials over
  \(\R\) of degree less than \(n\).
\begin{enumerate}[label=(\roman*)]
  \item Show that the map \(T\colon\mathcal{P}_3\to\mathcal{P}_4\) given
    by
    \[
    T(p(x))=6\int_1^x p(t)\,dt
    \]
    is linear. Determine whether it is 1--1 or onto.
  \item Let \(\mathcal{B}_3=\{1,x,x^2\}\) be a basis for \(\mathcal{P}_3\) and let
    \(\mathcal{B}_4=\{1,x,x^2,x^3\}\) be a basis for \(\mathcal{P}_4\). Find the
    matrix representation of \(T\) with respect to these bases. \\
    \end{enumerate}
\end{exercise}

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

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

% HW 12, Problem 2
\begin{exercise}
Let \(T\colon X\to U\), with \(\dim X=n\) and \(\dim U=m\).  Show how
  to construct bases \(\mathcal{B}_X\) for \(X\) and \(\mathcal{B}_U\) for \(U\)
  such that the matrix of \(T\) in block form is
  \[
  M=\begin{bmatrix}I_r&0\\0&0\end{bmatrix},
  \]
  where \(I_r\) is the \(r\times r\) identity matrix, and the other blocks
  are either empty or contain all zeros. Prove/justify all of your claims. \\
\end{exercise}

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

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

% HW 12, Problem 3
\begin{exercise}
Let \(X\) be a vector space with basis \(x_1,x_2,x_3\), and consider the
  linear map \(T:X\to X\) with matrix representation
  \(\begin{bmatrix}1&-1&0 \\ 0&2&-2 \\ -3&0&3\end{bmatrix}\) with
    respect to this basis. What is the matrix
    representation of \(T\) with respect to the basis \(x_1-x_2\),
    \(x_2-x_3\), \(x_1+x_3\).
\end{exercise}

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

%%--------------------------------------------------------------------- 
    
\end{document}