Affine space - Wikipedia
Affine space is the setting for affine geometry. As in Euclidean space, the fundamental objects in an affine space are called points, which can be thought of as locations in the space without any size or shape: zero- dimensional.
Affine space - Encyclopedia of Mathematics
2020年4月4日 · In algebraic geometry an affine algebraic set is sometimes called an affine space. A finite-dimensional affine space can be provided with the structure of an affine variety with the Zariski topology (cf. also Affine scheme). Affine spaces associated with a vector space over a skew-field $ k $ are constructed in a similar manner. References
intuition - What is the affine space and what is it for?
An affine space is an abstraction of how geometrical points (in the plane, say) behave. All points look alike; there is no point which is special in any way. You can't add points. However, you can subtract points (giving a vector as the result).
Affine Space -- from Wolfram MathWorld
4 天之前 · In an affine space, it is possible to fix a point and coordinate axis such that every point in the space can be represented as an -tuple of its coordinates. Every ordered pair of points and in an affine space is then associated with a vector .
What are differences between affine space and vector space?
A vector space is an algebraic object with its characteristic operations, and an affine space is a group action on a set, specifically a vector space acting on a set faithfully and transitively. Why do we say that the origin is no longer special in the affine space?
affine space in nLab - ncatlab.org
2025年1月13日 · Clearly every vector space has an underlying affine space (and every linear map is affine linear), giving a forgetful functor U: Vect → Aff U:Vect \to Aff. Conversely, any affine space gives rise to a canonical vector space, sometimes called its space of displacements.
Affine space - GeeksforGeeks
2021年9月21日 · Affine space is the set E with vector space \vec{E} and a transitive and free action of the additive \vec{E} on set E. The elements of space A are called points. The vector space \vec{E} that is associated with affine space is known as free vectors and the action +: E * \vec{E} \rightarrow E satisfying the following conditions:
Affine transformation - Wikipedia
In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
2019年5月31日 · vectors in T of the form d(x,y) form a vector space. Notice that an affine space consists of a set X, vector space T, and difference function d; however, an affine space (per se) is not itself a vector space. Examples. With X = R (as a point set) and T = R1 (as a vector space), we can define d(x,y) = y − x.
Affine Spaces | Brilliant Math & Science Wiki
An affine space is a triple (A, V, +) (A,V,+) where A A is a set of objects called points and V V is a vector space with the following properties: a = b + \vec {v} a = b+v. It is apparent that the additive group V V induces a transitive group action upon A A; this directly follows from the definition of …
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