Hyperplane/Kernel of a linear form/Fact/Proof

Proof

Due to fact, there exists a one-dimensional linear subspace ${}W\subseteq V$ such that

${}V=U\oplus W\,.$

This decomposition determines a projection

$p_{W}\colon V\longrightarrow W,$

which is linear and has ${}U$ as its kernel. Since ${}W$ is one-dimensional, there exists an isomorphism

$\psi \colon W\longrightarrow K.$

Therefore, ${}\psi \circ p_{W}$ is a linear form, whose kernel is just ${}U$.

To fact