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\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due September 22
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2023 $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-3mm}

\noindent\emph{Read}: Lax, Chapter 3, pages 25--28, and Chapter 4,
pages 32--43. \medskip

%Throughout, $X$ is assumed to be a vector space of dimension $n<\infty$.

\begin{enumerate}

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

\item Show that whenever meaningful, 
  \begin{enumerate}
  \item $(ST)'=T'S'$
  \item $(T+R)'=T'+R'$
  \item $(T^{-1})'=(T')^{-1}$.
  \end{enumerate}
  Here, $S'$ denotes the transpose of $S$. Carefully describe what you
  mean by ``whenever meaningful'' in each case. \\
  
%%------------------------------------------------------------------------

\item Both of the equalities in this problem following easily from
  $N_{T'}=R_T^\perp$, either by applying it to the transpose map, or
  taking the annihilator of both sides of this equation.  Give an
  alternate direct algebraic proof of each, using the definitions. As
  usual, you should canonically identity $X''$ with $X$.
  \begin{enumerate}
  \item $N_T=R_{T'}^\perp$
  \item $N_{T'}^\perp=R_T$. \\
  \end{enumerate}

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

\item Let $\mathcal{P}_n$ be the vector space of all polynomials over
  $\R$ of degree less than $n$.
\begin{enumerate}
  \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 $\BBB_3=\{1,x,x^2\}$ be a basis for $\mathcal{P}_3$ and let
    $\BBB_4=\{1,x,x^2,x^3\}$ be a basis for $\mathcal{P}_4$. Find the
    matrix representation of $T$ with respect to these bases. \\
\end{enumerate}

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

\item Let $T\colon X\to U$, with $\dim X=n$ and $\dim U=m$.  Show how
  to construct bases $\BBB_X$ for $X$ and $\BBB_U$ 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. \\

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

\item 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{enumerate}

\end{document}
