%% LyX 2.3.3 created this file. For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass[oneside,english]{amsbook}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\usepackage{amsthm}
\makeatletter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
\numberwithin{section}{chapter}
\numberwithin{equation}{section}
\numberwithin{figure}{section}
\newenvironment{lyxlist}[1]
{\begin{list}{}
{\settowidth{\labelwidth}{#1}
\setlength{\leftmargin}{\labelwidth}
\addtolength{\leftmargin}{\labelsep}
\renewcommand{\makelabel}[1]{##1\hfil}}}
{\end{list}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\include{mydefs}
\usepackage{fullpage}
\makeatother
\usepackage{babel}
\begin{document}
\section*{Lior Silberman's Math 223: Problem Set 5 (due 22/2/2021)}
\begin{center}
\textbf{Calculations with matrices}
\par\end{center}
\begin{lyxlist}{10.}
\item [{1.}] Let $A=\left(\begin{array}{cc}
-2 & 3\\
5 & -7
\end{array}\right)$, $B=\left(\begin{array}{ccc}
4 & 1 & 0\\
0 & -2 & 9
\end{array}\right)$, $C=\left(\begin{array}{cc}
0 & 0\\
1 & 1\\
2 & 2
\end{array}\right)$, $D=\left(\begin{array}{ccc}
7 & 0 & 0\\
0 & 6 & 0\\
0 & 0 & 5
\end{array}\right)$. Calculate all possible products among pairs of $A,B,C,D$ (don't
forget that $A^{2}=AA$ is also such a product and that $XY,YX$ are
different products if both make sense).\bigskip{}
\item [{PRAC}] The \emph{$n\times n$ identity matrix} is the matrix $I_{n}\in M_{n}(\R)$
with entries: $\left(I_{n}\right)_{ij}=\begin{cases}
1 & i=j\\
0 & i\neq j
\end{cases}$. Show that $I_{n}\underline{v}=\underline{v}$ for all $\underline{v}\in\R^{n}$.\bigskip{}
\item [{2.}] Let $A\in M_{m,n}(\R)$. Show that $AI_{n}=I_{m}A=A$. (Hint)\bigskip{}
\item [{PRAC}]~
\begin{lyxlist}{10.}
\item [{(a)}] Let $A\in M_{n,m}(\R)$, $B\in M_{m,p}(\R)$. Show that the
$j$th column of $AB$ is given by the product $A\underline{v}$ where
$\underline{v}$ is the $j$th column of $B$.
\item [{(b)}] Let $A\in M_{n,m}(\R)$, $B\in M_{m,p}(\R)$. Show that the
$j$th column of $AB$ is a linear combination of all the columns
of $A$ with the coefficients being the $j$th column of $B$.\bigskip{}
\end{lyxlist}
\item [{3.}] Let $A,B\in M_{n}(\R)$ be square matrices. We say $A,B$
\emph{commute} if $AB=BA$. WE say $A$ is \emph{scalar }if $A=zI_{n}$
for some $z\in\R$. The \emph{centre} of $M_{n}(\R)$ is the set $Z=\left\{ A\in M_{n}(\R)\mid\forall B\in M_{n}(\R):AB=BA\right\} $
of matrices that commute with all other matrices.
\begin{lyxlist}{10.}
\item [{PRAC}] Check that the action of $zI_{n}$ on vectors is by multiplication
by the scalar $z$.
\item [{(a)}] Show that $Z\subset M_{n}(\R)$ is a subspace.
\item [{(b)}] Show that the centre of $M_{n}(\R)$ consists of scalar matrices:
$Z=\Span_{\R}(I_{n})$.\bigskip{}
\end{lyxlist}
\item [{4.}] Let $A=\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)\in M_{2}(\R)$ and suppose that $ad-bc\neq0$.
\begin{lyxlist}{10.}
\item [{(a)}] Find a matrix $B=\left(\begin{array}{cc}
e & f\\
g & h
\end{array}\right)$ such that $AB=I_{2}=\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right)$. Show that $BA=I_{2}$ as well.
\item [{({*}b)}] (``Uniqueness of the inverse'') Suppose that $AC=I_{2}$.
Show that $C=B$.\bigskip{}
\end{lyxlist}
\item [{{*}5.}] Find a matrix $N\in M_{2}(\R)$ such that $N^{2}=0$ but
$N\neq0$.\bigskip{}
\item [{6.}] (``Group homomorphisms'')
\begin{lyxlist}{10.}
\item [{(a)}] Let $R_{\alpha}$ be the matrix $\left(\begin{array}{cc}
\cos\alpha & -\sin\alpha\\
\sin\alpha & \cos\alpha
\end{array}\right)$ (``rotation in the plane by angle $\alpha$''). Show that $R_{\alpha}R_{\beta}=R_{\alpha+\beta}$.
\item [{(b)}] Let $n(x)$ be the matrix $\left(\begin{array}{cc}
1 & x\\
0 & 1
\end{array}\right)$ (``shear in the plane by $x$''). Show that $n(x)n(y)=n(x+y)$.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{An application to graph theory}
\par\end{center}
\begin{lyxlist}{10.}
\item [{{*}7.}] Let $V$ be a vector space. A linear map $T\colon V\to V$
is said to be \emph{bipartite} if there are subspaces $W_{1},W_{2}\subset V$
such that $V=W_{1}\oplus W_{2}$ (internal direct sum). and such that
$T(W_{1})\subset W_{2}$ and $T(W_{2})\subset W_{1}$. Let $T$ be
bipartite with respect to the decomposition $V=W_{1}\oplus W_{2}$.
Show that $\dim\Ker T\geq\left|\dim W_{1}-\dim W_{2}\right|$.\bigskip{}
\end{lyxlist}
Hint for 2: interpret the compositions as linear maps, and use the
practice problem.
Hint for 3a: use the practice problem and a previous problem set.\bigskip{}
\begin{center}
\textbf{Supplementary problems}
\par\end{center}
\begin{lyxlist}{10.}
\item [{A.}] Show by hand that for any three matrices $A,B,C$ with compatible
dimensions, $(AB)C=A(BC)$.
\item [{B.}] (Every vector space is $\R^{n}$) Let $V$ be a vector space
with basis $B=\left\{ \underline{v}_{i}\right\} _{i\in I}$ ($I$
may be infinite).
\begin{lyxlist}{10.}
\item [{(a)}] Let $\Phi\colon\R^{\oplus I}\to V$ be the map $\Phi(f)=\sum_{i\in I}f_{i}\underline{v}_{i}=\sum_{f_{i}\neq0}f_{i}\underline{v}_{i}$
{[}recall that we admit infinite sums where only finitely many summands
are non zero{]}. Show that $\Phi$ is a an isomorphism of vector spaces.
\item [{RMK}] The inverse map $\Psi\colon V\to\R^{\oplus I}$ is called
the \emph{coordinate map} (in the ordered basis $B$)
\item [{(b)}] Construct an isomorphism $V^{*}\to\R^{I}$.
\item [{(c)}] Let $W$ be another space with basis $C=\left\{ \underline{w}_{j}\right\} _{j\in J}$.
Construct an injective linear map $\Hom(V,W)\to M_{I\times J}(\R)=\R^{I\times J}$
and show that its image is the set of matrices having at most finitely
many non-zero entries in each column.
\end{lyxlist}
\end{lyxlist}
\end{document}