rank of a linear mappingRankLinearMapping2002-02-19 18:49:582007-01-18 05:06:54Definition\usepackage{amsmath}
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\newcommand{\rank}{\operatorname{rank}}The \emph{rank} of a linear mapping $L\colon U\to V$ is defined to be
the $\dim L(U)$, the dimension of the mapping's image. Speaking less
formally, the rank gives the number of independent linear constraints
on $u\in U$ imposed by the equation
\[ L(u)=0. \]
\subsubsection*{Properties}
\begin{enumerate}
\item If $V$ is finite-dimensional, then $\rank L=\dim V$ if and only
if $L$ is surjective.
\item If $U$ is finite-dimensional, then $\rank L=\dim U$ if and only
if $L$ is injective.
\item Composition of linear mappings does not increase rank. If
$M\colon V\to W$ is another linear mapping, then \[\rank ML \le
\rank L\] and
\[\rank ML \le \rank M.\] Equality holds in the first case if
and only if $M$ is an isomorphism, and in the second case if and
only if $L$ is an isomorphism.
\end{enumerate}