\begin{definition}[Definition:Set of All Linear Transformations/Vector Space]
Let $K$ be a field.
Let $X, Y$ be vector spaces over $K$.
Then $\map {\LL} {X, Y}$ is the '''set of all linear transformations''' from $X$ to $Y$:
:$\map {\LL} {X, Y} := \set {\phi: X \to Y: \phi \mbox{ is a linear transformation} }$
\end{definition}