image of a linear transformationImageOfALinearTransformation2003-07-29 02:25:132004-09-24 16:41:38Definition% this is the default PlanetMath preamble. as your knowledge
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{\bf Definition}
Let $T:V\to W$ be a linear transformation. Then the {\bf image} of
$T$ is the set
$$ \operatorname{Im} (T) = \{ w\in W \mid w=T(v) \,\mbox{for some}\, v\in V\} = T(V).$$
\subsubsection{Properties}
%Let $T$ be as above.
\begin{enumerate}
\item The dimension of $\operatorname{Im}(T)$ is called the rank of $T$;
\item $T$ is a surjection, if and only if $\operatorname{Im}(T)=W$;
\item $\operatorname{Im}(T)$ is a vector subspace of $W$;
\item If $L\colon W\to U$ is a linear transformation, then $\operatorname{Im}(LT) =L(\operatorname{Im}(T))$;
\end{enumerate}