\begin{definition}[Definition:Kernel of Linear Transformation/Vector Space]
Let $\struct {\mathbf V, +, \times}$ be a vector space.
Let $\struct {\mathbf V', +, \times}$ be a vector space whose zero vector is $\mathbf 0'$.
Let $T: \mathbf V \to \mathbf V'$ be a linear transformation.
Then the '''kernel''' of $T$ is defined as:
:$\map \ker T := T^{-1} \sqbrk {\set {\mathbf 0'} } = \set {\mathbf x \in \mathbf V: \map T {\mathbf x} = \mathbf 0'}$
\end{definition}