linear isomorphism
LinearIsomorphism
2004-09-17 13:23:34
2009-03-30 17:30:06
Definition
isomorphism
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\begin{defn}
Suppose $V$ and $W$ are vector spaces and $L\colon V\to W$ is a linear map. Then $L$ is a \emph{linear isomorphism} if $L$ is bijective.
\end{defn}
\subsubsection*{Properties}
\begin{enumerate}
\item Compositions and of linear isomorphisms is a linear isomorphism.
\item The inverse of a linear isomorphisms is a linear isomorphism.
\item If either $V$ or $W$ if finite dimensional, then $\dim V=\dim W$.
(This is a consequence of the rank-nullity theorem.)
\end{enumerate}