Linear variety

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linear manifold, affine subspace

A subset $M$ of a (linear) vector space $E$ that is a translate of a linear subspace $L$ of $E$, that is, a set $M$ of the form $x_0 + L$ for some $x_0$. The set $M$ determines $L$ uniquely, whereas $x_0$ is defined only modulo $L$: $$ x_0 + L = x_1 + N $$ if and only if $L = N$ and $x_1 - x_0 \in L$. The dimension of $M$ is the dimension of $L$. A linear variety corresponding to a subspace of codimension 1 is called a hyperplane.

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Linear variety. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article