Linear mapping/Fiber/Affine subspace/Kernel/Example

For a linear mapping

$\varphi \colon V\longrightarrow W$

between ${}K$-vector spaces ${}V$ and ${}W$ and an element ${}Q\in W$, the preimage for ${}Q$ (the fiber over ${}Q$)

${}\varphi ^{-1}(Q)={\left\{P\in V\mid \varphi (P)=Q\right\}}\,$

is an affine subspace of ${}V$. If this is non-empty, then we can take any point ${}P_{0}\in V$ with

${}\varphi (P_{0})=Q\,$

as starting point The translation space is the kernel of ${}\varphi$. When a linear mapping is given, then ${}V$ is partitioned in a layered family of parallel affine subspaces.