Proposition: Quotient Space

Let \(V\) be a vector space over a field \(F\), and let \(U\subseteq V\) be its subspace. Let \(V/_U\) be the quotient. set of all equivalent classes defined by the equivalence relation induced by \(U\) on \(V\). and let $q\colon V\rightarrow V/U,\,v\mapsto [v]\,$ be its canonical projection. Then, there exists a uniquely defined vector space on \(V/_U\), called quotient space such that \(q\) is a linear map over the field \(F\).

Proofs: 1

Definitions: 1

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Adapted from CC BY-SA 3.0 Sources:

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