15-00-ZeroVectorSpace.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ZeroVectorSpace}
\pmcreated{2013-03-22 14:03:32}
\pmmodified{2013-03-22 14:03:32}
\pmowner{drini}{3}
\pmmodifier{drini}{3}
\pmtitle{zero vector space}
\pmrecord{6}{35414}
\pmprivacy{1}
\pmauthor{drini}{3}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{15-00}
\pmsynonym{trivial vector space}{ZeroVectorSpace}
\pmrelated{ZeroRing}
\pmrelated{ZeroMap}

\endmetadata

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\newcommand{\sC}[0]{\mathbb{C}}
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\newcommand{\sZ}[0]{\mathbb{Z}}

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\begin{document}
{\bf Definition}
A {\bf zero vector space} is a vector space that contains only one element, a zero vector. 

\subsubsection{Properties}
\begin{enumerate}
\item Every vector space has a zero vector space as  a vector subspace.
\item A vector space $X$ is a zero vector space if and only if the dimension of $X$ is
zero.
\item Any linear map defined on a zero vector space is the zero map. 
If $T$ is linear on $\{0\}$, then $T(0)=T(0\cdot 0) = 0T(0)=0$. 
\end{enumerate}
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\end{document}