Vector/Linear form/Zero test/Fact/Proof

Proof

The one-dimensional ${}K$-linear subspace ${}Kv\subseteq V$ contains a direct complement, that is,

${}V=Kv\oplus U\,$

with some linear subspace ${}U\subseteq V$. The projection onto ${}Kv$ for this decomposition sends ${}v$ to ${}1$.

To fact