(related to Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace)
We want to show that if \(V\) is a vector space over a field \(F\) and if \(U\subseteq V\) is its subspace, then the relation defined by \(v\sim w \Longleftrightarrow v-w\in U,\) defines an equivalence relation on \(V\). We have to check all properties of the equivalence relation: