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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.
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".
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
The affine hull is the smallest affine subspace of a Euclidean space containing a given set, or the union of all affine combinations of points in the set. [39] The linear hull is the smallest linear subspace of a vector space containing a given set, or the union of all linear combinations of points in the set. [39]
In particular, any affine combination of the fixed points of a given affine transformation is also a fixed point of , so the set of fixed points of forms an affine space (in 3D: a line or a plane, and the trivial cases, a point or the whole space).
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 ...
For example, the set of all vectors (x, y, z) (over real or rational numbers) satisfying the equations + + = + = is a one-dimensional subspace. More generally, that is to say that given a set of n independent functions, the dimension of the subspace in K k will be the dimension of the null set of A , the composite matrix of the n functions.