15-00-ZeroMap.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ZeroMap}
\pmcreated{2013-03-22 14:03:38}
\pmmodified{2013-03-22 14:03:38}
\pmowner{matte}{1858}
\pmmodifier{matte}{1858}
\pmtitle{zero map}
\pmrecord{6}{35416}
\pmprivacy{1}
\pmauthor{matte}{1858}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{15-00}
\pmrelated{ZeroVectorSpace}
\pmrelated{ConstantFunction}
\pmrelated{IdentityMap}
\pmdefines{zero operator}

\endmetadata

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\begin{document}
{\bf Definition}
Suppose $X$ is a set, and $Y$ is a vector space with zero vector $0$. 
If $Z$ is a map $Z:X\to Y$, such that $Z(x)=0$ for all $x$ in $X$,
then $Z$ is a {\bf zero map}.

\subsubsection{Examples}
\begin{enumerate}
\item On the set of non-invertible $n\times n$ matrices, the determinant
is a zero map.
\item If $X$ is the zero vector space, any linear map $T:X\to Y$ is
a zero map. In fact, $T(0)=T(0\cdot 0)=0T(0)=0$. 
\item If $X=Y$ and its field is $\R$ or $\C$, then the spectrum of $Z$ is
$\{0\}$. 
\end{enumerate}
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