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\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due February 14
    (Tuesday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Spring 2017 $|$ M.~Macauley}}


\begin{document}

$\;$

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

\begin{enumerate}


  % Lax Exercise 3.10, 3.11
\item Let $A,B\colon X\to X$ be linear maps.
  \begin{enumerate} 
  \item Show that if $A$ is invertible and similar to $B$, then $B$ is
    also invertible, and $B^{-1}$ is similar to $A^{-1}$.
  \item Show that if either $A$ or $B$ is invertible, then $AB$ and
    $BA$ are similar. \\
  \end{enumerate}



\item Let $T:X\to X$ be linear, with $\dim X=n$.
  \begin{enumerate}
  \item Prove that if $T^2=T$, then $X=R_T\oplus N_T$. 
  \item Show by example that if $T^2\neq T$, then $X=R_T\oplus N_T$
    need not hold.
  \item Prove that $N_{T^n}=N_{T^{n+1}}$ and
    $R_{T^n}=R_{T^{n+1}}$. 
  \item Prove that $X=R_{T^n}\oplus N_{T^n}$.
  \item Show there exists a linear map $S:X\to X$ such that $ST=TS$
    and $ST^{n+1}=T^n$. \\
  \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. Indicate 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 that
  there exist bases $\BBB$ for $X$ and $\BBB'$ for $U$ such that the matrix
  of $T$ in block form is
  \[
  M=\begin{bmatrix}I_k&0\\0&0\end{bmatrix}\,
  \]
  where $I_k$ is the $k\times k$ identity matrix, and the other blocks
  are either empty or contain all zeros. \\



 \item Consider the linear map $T:\R^3\to\R^3$ with matrix
   representation $\begin{bmatrix}1&-1&0 \\ 0&2&-2
     \\ -3&0&3\end{bmatrix}$ with respect to the standard basis. What
     is the matrix representation of $T$ with respect to the basis
     $\{(1,-1,0),(0,1,-1),(1,0,1)\}$? 



\end{enumerate}

\end{document}
