| Version current |
Version 2 |
| {\bf Definition} |
{\bf Definition} |
| A {\bf zero vector space} is a vector space that contains only one element, a zero vector. |
A {\bf zero vector space} is a vector space that contains only one element, a zero vector. |
|
|
| \subsubsection{Properties} |
\subsubsection{Properties} |
| \begin{enumerate} |
\begin{enumerate} |
| \item Every vector space has a zero vector space as a vector subspace. |
\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 |
\item A vector space $X$ is a zero vector space if and only if the dimension of $X$ is |
| zero. |
zero. |
| \item Any linear map defined on a zero vector space is the zero map. |
\item Any linear map defined on a zero vector space is the zero map. |
| If $T$ is linear on $\{0\}$, then $T(0)=T(0\cdot 0) = 0T(0)=0$. |
If $T$ is linear on $\{0\}$, then $T(0)=T(0\cdot 0) = 0T(0)=0$. |
| \end{enumerate} |
\end{enumerate} |
