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Origins from Alice's and Bob's perspectives. Vector computation from Alice's perspective is in red, whereas that from Bob's is in blue. The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after one has forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, "An ...
Informally, an affine space is a vector space without a fixed choice of origin. It describes the geometry of points and free vectors in space. As a consequence of the lack of origin, points in affine space cannot be added together as this requires a choice of origin with which to form the parallelogram law for vector addition.
In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line.
Affine geometry can be developed in two ways that are essentially equivalent. [3] In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom). Affine geometry can also be developed on the basis of linear algebra.
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 particular, every linear space is also an affine space.
By Hilbert's basis theorem and the fact that Noetherian rings are closed under quotients, every affine or projective coordinate ring is Noetherian. As a consequence, affine or projective spaces with the Zariski topology are Noetherian topological spaces, which implies that any closed subset of these spaces is compact.
As affine spaces can be embedded in projective spaces, all affine varieties can be embedded in projective spaces too. Any choice of a finite system of nonsimultaneously vanishing global sections of a globally generated line bundle defines a morphism to a projective space.
In more fancy terms, affine morphisms are defined by the global Spec construction for sheaves of O X-Algebras, defined by analogy with the spectrum of a ring. Important affine morphisms are vector bundles, and finite morphisms. 5. The affine cone over a closed subvariety X of a projective space is the Spec of the homogeneous coordinate ring of X.