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An affine space is a subspace of a projective space, which is in turn the quotient of a vector space by an equivalence relation (not by a linear subspace) Affine spaces are contained in projective spaces. For example, an affine plane can be obtained from any projective plane by removing one line and all the points on it, and conversely any ...
In geometry, a flat is an affine subspace, i.e. a subset of an affine space that is itself an affine space. [1] Particularly, in the case the parent space is Euclidean, a flat is a Euclidean subspace which inherits the notion of distance from its parent space.
Here is an example in algebraic geometry. Let U be an open affine chart on a projective variety X (in the Zariski topology). Then each closed subvariety Y of U is locally closed in X ; namely, Y = U ∩ Y ¯ {\displaystyle Y=U\cap {\overline {Y}}} where Y ¯ {\displaystyle {\overline {Y}}} denotes the closure of Y in X .
Linear subspace, in linear algebra, a subset of a vector space that is closed under addition and scalar multiplication; Flat (geometry), a Euclidean subspace; Affine subspace, a geometric structure that generalizes the affine properties of a flat; Projective subspace, a geometric structure that generalizes a linear subspace of a vector space
An affine space need not be included into a linear space, but is isomorphic to an affine subspace of a linear space. All n -dimensional affine spaces over a given field are mutually isomorphic. In the words of John Baez , "an affine space is a vector space that's forgotten its origin".
In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V.The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can ...
A flat, Euclidean subspace or affine subspace of E is a subset F of E such that ... For example, the Cartesian coordinates of a vector on an ...
[1]: 3 An irreducible affine algebraic set is also called an affine variety. [1]: 3 (Some authors use the phrase affine variety to refer to any affine algebraic set, irreducible or not. [note 1]) Affine varieties can be given a natural topology by declaring the closed sets to be precisely the affine algebraic sets. This topology is called the ...