\magnification 1800
\parskip 10pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein

\def\R{{\bf R}}
\def\C{{\bf C}}
\def\x{{\bf x}}
\def\r{{\bf r}}
\def\c{{\bf c}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\a{{\bf a}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\p{{\bf p}}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\u{\vskip -8pt}

\def\N{{\bf N}}
\def\S{\Sigma_{i=1}^n}



2.7 \hfil \break
 Let $A_{m \times n} = \left(\matrix{\r_1 \cr \r_2 \cr ... \cr \r_m }\right)$
$= \left(\matrix{\c_1 & \c_2 & ... & \c_n }\right)$


Rank of $A$ = $dim(span\{\r_1, ..., \r_m\}) = dim(span\{\c_1, ..., \c_m\})$ 

\rightline{= maximum order of any nonvanishing minor determinant.}

Ex:  $\left(\matrix{1 & 2 & 3 & 4 \cr 0 & 0 & 0 & 0 \cr 0 & 0 & 5 & 6  }\right)$
$\sim \left(\matrix{1 & 3 & 2 & 4 \cr 0 & 0 & 0 & 0 \cr 0 & 5 & 0 & 6  }\right)$



Let $F:  U \subset \R^n \rightarrow \R^m \in C^1$.

Rank of $F$ at $x$ = rank of $DF(x)$ 

$F$ has rank $k$ if $F$ has rank $k$ at each $x$.

$Det:  M^{n \times m} \rightarrow \R$ is a continuous function.


Suppose rank $DF(a) = k$  implies there exists $V$ open such that $a \in V$ and $DF(x) \geq k$ for all $x \in V$

Ex:   $F(x_1, x_2) = (x_1x_2 + 5, x_1 + x_2 - 3)$

$DF = \left(\matrix{x_2 & x_1 \cr  1 & 1 }\right)$

\eject

Rank Theorem:  Suppose $A_0 \subset \R^n$, $B_0 \subset \R^m$, $F: A_0 \rightarrow B_0 \in C^1$ 
$a \in A_0, b \in B_0$.  Suppose rank F = $k$.

Then there exists $A^{open} \subset A_0$ such that $a \in A$ and  $B^{open} \subset B_0$ such that $b \in B$ and $G, H$, $C^r$ diffeomorphisms

such that $G:  A \rightarrow U^{open} \subset \R^n$, $H:  B \rightarrow V^{open} \subset \R^m$ and
$$H \circ F \circ G^{-1} (x_1, ..., x_n) = (x_1, ..., x_k, 0, ..., 0)$$



\end