Affine subspace

Definition

Let $K$ be a field, let $V$ be a vector space over $K$ , and let $A$ be an affine space over $V$ . A nonempty subset $S$ of $A$ is an affine subspace if $S = p + W$ for a point $p$ of $A$ and a linear subspace $W$ of $V$ . This means that $S = p + W = \{\, p + w \;:\; w \in W \,\}.$ The subspace $W$ is the direction of $S$ . It is unique: $W = \{\, q - r : q, r \in S \,\}$ . The dimension of $S$ is the dimension of $W$ .

Remarks

For each point $p$ of $S$ , $S = p + W$ . Thus each point of $S$ can serve as the base point. In $\mathbb{R}^n$ , the affine subspaces of dimension 1 are the lines, and those of dimension 2 are the planes. Those of dimension $n - 1$ are the hyperplanes. Two affine subspaces are parallel if they have the same direction. Mathlib's AffineSubspace also allows the empty set. A subset $S$ is an affine subspace in this sense if $c(q - r) + s \in S$ for all $c \in K$ and $q, r, s \in S$ .

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Affine space concept
Affine subspace definition

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Lévy–Steinitz theorem

Tag	`0FBXFBZBH8`
URI	`m2://0FBXFBZBH8`
Kind	definition
Writing	W1 stated
MSC	51N10
Defines	Affine subspace
Uses	Affine space

Affine subspace

Definition

Remarks

Reading path

Where this leads

Further reading

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