Let $K$ be a field, let $V$ be a nonzero [vector space](/page/Vector%20Space) over $K$, and let $V_0 := V \setminus \{0\}$. Define a relation $\sim$ on $V_0$ by declaring that, for $v,w \in V_0$,
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\begin{align*}
v \sim w \iff w = \lambda v \text{ for some } \lambda \in K^\times.
\end{align*}
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Then $\sim$ is an [equivalence relation](/page/Equivalence%20Relation), and the map