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| Title of object: |
zero map |
| Canonical Name: |
ZeroMap |
| Type: |
Definition |
| Created on: |
2003-11-01 04:02:18 |
| Modified on: |
2003-11-01 04:02:18 |
| Classification: |
msc:15-00 |
| Defines: |
zero operator |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}
% The below lines should work as the command
% \renewcommand{\bibname}{References}
% without creating havoc when rendering an entry in
% the page-image mode.
\makeatletter
\@ifundefined{bibname}{}{\renewcommand{\bibname}{References}}
\makeatother
\newcommand*{\norm}[1]{\lVert #1 \rVert}
\newcommand*{\abs}[1]{| #1 |} |
Content:
{\bf Definition}
Suppose $X$ is a set, and $Y$ is a vector space with zero vector $0$.
If $T$ is a map $T:X\to Y$, such that $T(x)=0$ for all $x$ in $X$,
then $T$ 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, $T(0)=T(0\cdot 0)=0T(0)=0$.
\end{enumerate} |
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