zero mapZeroMap2003-11-01 04:02:182005-12-11 08:25:51Definitionzero operator% almost certainly you want these
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\newcommand*{\abs}[1]{| #1 |}{\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}