\documentclass[11pt]{article}
\usepackage{amsmath,amsthm,amsfonts,amssymb,amscd}
\usepackage{graphicx, epsfig}
\usepackage{multicol}
\usepackage{boxedminipage}
\usepackage{subfigure}
\usepackage{euler,palatino,array}
\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref}
\usepackage{wrapfig}
\usepackage{bbold}


\setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.5in} 
\setlength{\parindent}{0in}
\newcounter{prob}
\setcounter{prob}{0}
\newcommand{\problem}{\stepcounter{prob}\medskip{\bf Presentation problem \theprob.}\ }
\newcommand{\answer}{\medskip{\bf Answer to Question \theprob.}\ }
\newcommand{\solution}{\medskip{\bf Solution to Question \theprob.}\ }



% ***** Some useful abreviations *****
\newcommand{\rank}{\mathrm{rank}}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\K}{\mathbb{K}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\B}{\mathscr{B}}
\newcommand{\LL}{\mathscr{L}}
\newcommand{\PP}{\mathscr{P}}
\newcommand{\ra}{\rightarrow}
\newcommand{\ds}{\displaystyle}
\newcommand{\bnum}{\begin{enumerate}} %Use with \item to create numbered lists
\newcommand{\enum}{\end{enumerate}}
\newcommand{\babc}{\bnum\renewcommand{\labelenumi}{(\alph{enumi})}}%Use with \item for abc lists
\newcommand{\eabc}{\end{enumerate}}
\newcommand{\bijk}{\bnum\renewcommand{\labelenmi}{\roman{enumi}.}}%Roman numeral lists
\newcommand{\eijk}{\end{enumerate}}

\renewcommand{\ss}{\smallskip}
\newcommand{\ms}{\medskip}
\newcommand{\bs}{\bigskip}




\pagestyle{myheadings} \markright{Math 4310 (Fall 2020) \hspace{2in}
(HW5)}

\begin{document}


%% PREAMBLE

\thispagestyle{empty}

\begin{minipage}[t]{1.1in}
{\vspace{-0.2in}
  \includegraphics[width=1.1in]{cornell-logo.jpg}}
\end{minipage}
\hfill
\begin{minipage}[t]{3in}
%\vspace{.05in}
  {\large Homework \# 5 $\quad$ Math 4310 Fall 2020

    \medskip
    
    Due 10/9/20
  }

\end{minipage}
\hfill
\begin{minipage}[t]{3.4in}
\end{minipage}

\vskip 0.25in

%\vspace*{-66mm}


Please submit your completed homework via gradescope on canvas.
I encourage you to work with your classmates on this homework (except for the journal entries!)
When you submit your work, please {\bf list your collaborators}.
(Your grade will not be affected.)
Even if you work in a group, you should write up 
your solutions {\bf yourself}! 
You should include all computational details, and proofs should be carefully 
written with full details.
As always, please write {\bf neatly and legibly} (feel free to use \LaTeX\ to write up your solutions, if you wish!).

\vskip 4mm

%% END PREAMBLE

\noindent {\bf Journal entry.}
How are you finding this class?  Specifically, are we covering the material at a speed that is reasonable for you?
What do you think of zoom lectures?  Any suggestions for me to improve the course?

\vskip 0.1in

\noindent {\bf Exercises.} 

\vskip 0.1in

\begin{enumerate}

    \item Let $T\in \mathscr{L}(\R^3,\R^3)$ and suppose
$$
T\left(\left[\begin{array}{r}1\\0\\-1\end{array}\right]\right)=\left[\begin{array}{r}1\\1\\1\end{array}\right] \ , \
T\left(\left[\begin{array}{r}0\\1\\-1\end{array}\right]\right)=\left[\begin{array}{r}1\\1\\0\end{array}\right] \ , \mbox{ and }
T\left(\left[\begin{array}{r}0\\0\\1\end{array}\right]\right)=\left[\begin{array}{r}1\\0\\0\end{array}\right].
$$
Let $\mathcal{S}$ denote the standard basis $e_1,e_2,e_3$.  Determine $[T]_{\mathcal{S}}$.

For the basis $\mathcal{B} = \left\{ \left[\begin{array}{r}1\\0\\-1\end{array}\right],\left[\begin{array}{r}0\\1\\-1\end{array}\right],
\left[\begin{array}{r}0\\0\\1\end{array}\right]\right\}$, determine $[T]_{\mathcal{B}}$.

\item We will let $\mathscr{L}(V)$ denote all linear maps from a vector space $V$ to itself (sometimes called
{\bf linear operators}.).  Let $T\in\mathscr{L}(V)$, and let $T^2$ denote the composition $T\circ T$.
\babc
%\item Give an example of a vector space $V$ and a linear operator $T\in\mathscr{L}(V)$, with $T\neq \mathbb{0}$ and
%$T\neq \mathbb{1}_V$, so that $T^2 = T$.

\item Prove that if $T^2=T$, then $V=\ker(T)\oplus \ker(T-\mathbb{1}_V)$.  

% This problem relies on the silly trick v = v-T(v)+T(v).......

\item Prove that if $V = \ker(T)+\ker(T-\mathbb{1}_V)$, then $T^2=T$.

\item Give an example of a vector space $V$ and $T\in\mathscr{L}(V)$ such that
$T^2 = -\mathbb{1}_V$.

\item Prove that if $T^2=\mathbb{0}_{V\to V}$ is the zero map, then $\rank(T)\leq \frac{\dim(V)}{2}$.

\eabc

\item Suppose that $c_0, \ldots, c_n$ are distinct elements of a field $\F$.  Show that the linear map
  \[ T : \PP_n(\F) \longrightarrow \F^{n+1} \]
  defined by \[T(f(x)) = \begin{pmatrix} f(c_0) \\ f(c_1) \\ \vdots \\ f(c_n) \end{pmatrix}\]
  is an isomorphism. (Hint: recall Lagrange interpolation).
  
\item Is the following linear map $T$ invertible?  If so, find its inverse: $T : \R^3 \longrightarrow \R^3$, given by $T(a,b,c) := (3a+2c, b, a+2b+c)$.
  Justify your answers!

\item Let $a, b, c, d \in \R$ be real numbers. Let
  \[
  B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.
  \]
  Consider the linear map
  $T : \R^{2\times 2} \longrightarrow \R^{2\times 2}$ defined by $T(A) := BA$.
  Consider the basis ${\mathcal B} = (E_{11}, E_{12}, E_{21}, E_{22})$, of $\R^{2\times 2}$ where
  $E_{ij}$ is the $2 \times 2$ matrix with a 1 in the $i$th row and $j$th column, and zeros elsewhere.
  \babc
\item  Find the matrix of $T$ with respect to ${\mathcal B}$ (your matrix will have $a, b, c, d$'s in it).
\item Find the rank of $T$, in terms of information (e.g. rank) of $B$.
    \eabc
    As always, justify your answers!
  
\item Suppose that $V, W, U$ are vector spaces, and that 
$T : V \longrightarrow W$ and $S : W \longrightarrow U$ are linear maps.
  \babc
\item Prove: if $ST$ is injective, then $T$ is injective.
\item Prove: if $ST$ is surjective, then $S$ is surjective.
  \item Prove: if $S$ is an isomorphism, then $rank(T) = rank(ST)$.
  \item Suppose that $S$ and $T$ are in $\mathscr{L}(V)$, where $V$ is finite dimensional.
    Show that $ST = \mathbb{1}_V$ if and only if $TS = \mathbf{1}_V$.

  \eabc

\item Let $A$ be an $n \times n$ matrix over $\R$, let $L_A : \R^n \longrightarrow \R^n$ denote the
  corresponding linear map.  Let $std$ denote
    the standard basis of $\R^n$, and let ${\mathcal B}$ be another basis
    for $\R^n$. Let $B = [L_A]_{\mathcal B}$ be the
    matrix of $L_A$ in the basis ${\mathcal B}$. In this problem, we try to relate the two matrices $A$ and $B$.
    Let $id : \R^n \longrightarrow \R^n$
    denote the identity map, and let $S = [id]_{std \leftarrow \mathcal B}$ be the matrix of $id$ with respect to these bases.
    It might help to recall from Friday's class (for $S$ and $T$ and bases as in class):
    \[ [ST]_{{\mathcal C} \leftarrow {\mathcal A}} = [S]_{{\mathcal C} \leftarrow {\mathcal B}} [T]_{{\mathcal B} \leftarrow {\mathcal A}}.
    \]
    \babc
    \item What is the matrix $[L_A]_{std}$?
    \item Show that $S^{-1} = [id]_{B \leftarrow std}$.
    \item  One of the following two statements holds,
      (1) $B = S^{-1} A S$, or (2) $B = S A S^{-1}$.  Which is correct? (Justify your answers!)
    \item Now we apply this to a specific matrix A:
      \[
      A = \begin{pmatrix} 1 & 1 & -1 \\ 2 & -1 & 0 \\ 1 & 0 & -1
        %A = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 1 & 0 & 1
      \end{pmatrix}.
      \]
  let ${\mathcal B}$ denote the following basis of $\R^3$:
  \[ \mathcal B = \left(\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix},
  \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix},
  \begin{pmatrix} 1 \\ 0 \\ 1\end{pmatrix}\right)
    \]
    \babc
  \item Find the matrix $B$ of $L_A$ in this basis
  \item Find the matrices $S$ and $S^{-1}$.
  \item Check your answer in (c) above on this example!  (You may do it by hand, or use software.  In the latter case, say what software you used, the commands you used to show this,
    and a screenshot or other rendition of the actual computation).
    \eabc
    \eabc
    

\end{enumerate}

\end{document}

