Definition: Affine Subspace

Let \(\mathcal A=(A,V_A,v)\) be an affine space. Let \(W\subseteq V_A\) is a subspace of the vector space \(V_A\) and let \(P\in\mathcal A\) be a point. A subset \(\mathcal U\subseteq \mathcal A\) is called an affine subspace of \(\mathcal A\), if: \[\mathcal U=\{Q\in \mathcal A\,:\,\overrightarrow{PQ}\in W\}=\{P+x\,|\,x\in W\}\]

i.e. if \(U\) consists of all points of \(\mathcal A\), which are translations of \(P\) by vectors \(x\) from the subspace \(W\subseteq V_A\).

Definitions: 1 2 3

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  1. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994