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Proof

(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:

• Reflexivity: \(v-v=0\in U\).
• Symmetry: \(w-v=-(v-w)\in U\).
• Transitivity: From \(u-v\in U\) and \(v-w\in U\) it follows \(u-w=(u-v)+(v-w)\in U\).
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References

Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück