\documentclass[12pt]{amsart}

\usepackage{241}

\title{21-241 Matrices and Linear Transformations \\
       Summer 1 2012 \\
       Course notes \\
       June 6}
\author{Paul McKenney}


\begin{document}

\maketitle

\section{Invertibility Revisited}

Recall the corollary to rank-nullity that we established;
\begin{cor}
  Let $A$ be an $m\times n$ matrix.  Then $\rank(A) + \nullity(A) = n$.
\end{cor}
[We also proved via rank-nullity that the column rank and row rank are the same, hence my usage of the generic term ``rank'' above.]

\begin{thm}
  Let $A$ be an $n\times n$ matrix.  Then the following are equivalent.
  \begin{enumerate}
    \item\label{inv}  $A$ is invertible.
    \item\label{basis}  The columns of $A$ form a basis of $\RR^n$.
    \item\label{soln}  For every $b\in\RR^n$, there is exactly one solution to the system $Ax = b$.
    \item\label{unique}  For every $b\in\RR^n$, there is at most one solution to the system $Ax = b$.
    \item\label{exists}  For every $b\in\RR^n$, there is at least one solution to the system $Ax = b$.
    \item\label{hom}  The only solution to $Ax = 0$ is $0$.
    \item\label{li}  The columns of $A$ are linearly independent.
    \item\label{span}  The columns of $A$ span $\RR^n$.
    \item\label{nul}  $\nullity(A) = 0$.
    \item\label{rank}  $\rank(A) = n$.
    \item\label{linv}  $A$ is left-invertible.
    \item\label{rinv}  $A$ is right-invertible.
    \item\label{inj}  $T_A$ is injective.
    \item\label{surj}  $T_A$ is surjective.
    \item\label{trans}  $A^\top$ is invertible.
  \end{enumerate}
\end{thm}

\begin{proof}
  We've seen that
  \[
    \eqref{unique}\iff\eqref{hom}\iff\eqref{li}\iff\eqref{nul}\iff\eqref{linv}\iff\eqref{inj}
  \]
  and
  \[
    \eqref{exists}\iff\eqref{span}\iff\eqref{rank}\iff\eqref{rinv}\iff\eqref{surj}
  \]
  We've also seen $\eqref{inv}\iff\eqref{soln}\iff\eqref{basis}$ and clearly $\eqref{basis}\implies \eqref{li},\eqref{span}$.  The rank-nullity theorem fills in the gap by proving $\eqref{nul}\iff\eqref{rank}$; for if $\nul(A) = 0$, then $\rank(A) = n - 0 = n$ by rank-nullity, and similarly if $\rank(A) = n$ then $\nul(A) = n - n = 0$.

  Finally, you've shown on your homework that $\eqref{inv}\iff\eqref{trans}$.
\end{proof}

\section{Example Problems}

\begin{example*}
  Calculate the rank and nullity of the matrix $A$ below.  Find bases for $\row(A)$, $\nul(A)$ and $\col(A)$.
  \[
    \begin{pmatrix}
      1 & 2 & 3 \\
      2 & 3 & 5 \\
      3 & 5 & 8 \\
      5 & 8 & 13 \\
    \end{pmatrix}
  \]
  What about $A^\top$?

  Note that $(13 21 34)$ is in $\row(A)$ and $(1,1,1)$ is in $\nul(A)$.  Find the unique linear combinations of your bases which produce these vectors.
\end{example*}

\begin{example*}
  Let $d_1,\ldots,d_n\in\RR$ be given.  Calculate the rank and nullity of $D = \diag(d_1,\ldots,d_n)$, and bases for $\row(D)$, $\nul(D)$ and $\col(D)$.
\end{example*}

We've been working exclusively with real numbers up to this point.  Soon we'll have to use complex numbers.  If I haven't said it before, let me say it now; everything we've done up to now works for complex numbers too.  Let's do some computations with those.  Recall the following fact about complex numbers;
\begin{fact}
  Let $z = a + bi$ be a nonzero complex number (so at least one of $a$ and $b$ is nonzero).  Then
  \[
    \frac{1}{z} = \frac{\bar{z}}{|z|^2} = \frac{a - bi}{a^2 + b^2}
  \]
  In particular if $z$ lies on the unit circle (so $z = \cos\theta + i\sin\theta$ for some $\theta$) then
  \[
    \frac{1}{z} = \bar{z}
  \]
\end{fact}
\begin{example*}
  Find the rank and nullity of the matrix $A$ below.  Find bases for $\row(A)$, $\nul(A)$ and $\col(A)$.
  \[
    \begin{pmatrix}
      1 + 2i & -1 & 0 \\
      1 - 2i &  i & 3 \\
        0    &  2 & 1 \\
    \end{pmatrix}
  \]
\end{example*}

\begin{example*}
  \begin{enumerate}
    \item  Let $p$ and $q$ be points in $\RR^2$ (ie, the plane).  Show that $\{p,q\}$ is linearly independent if and only if the points $0$, $p$, and $q$ are not colinear.
    \item  Let $p,q,r\in\RR^3$.  Show that $\{p,q,r\}$ is linearly independent if and only if the points $0$, $p$, $q$, and $r$ are not coplanar.
  \end{enumerate}
\end{example*}

\end{document}
