12 Facts About Affine space


In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

FactSnippet No. 1,513,039

The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace.

FactSnippet No. 1,513,040

Dimension of an affine space is defined as the dimension of the vector space of its translations.

FactSnippet No. 1,513,041

An affine space of dimension one is an affine line.

FactSnippet No. 1,513,042

An affine subspace of dimension in an affine space or a vector space of dimension is an affine hyperplane.

FactSnippet No. 1,513,043

Related searches

Euclidean space

An affine space is a set together with a vector space, and a transitive and free action of the additive group of on the set.

FactSnippet No. 1,513,044

The vector space is said to be associated to the affine space, and its elements are called vectors, translations, or sometimes free vectors.

FactSnippet No. 1,513,045

An affine transformation or endomorphism of an affine space is an affine map from that space to itself.

FactSnippet No. 1,513,046

Indeed, in most modern definitions, a Euclidean space is defined to be an affine space, such that the associated vector space is a real inner product space of finite dimension, that is a vector space over the reals with a positive-definite quadratic form .

FactSnippet No. 1,513,047

An or barycentric frame of an affine space is a generating set that is independent .

FactSnippet No. 1,513,048

Equivalently, } is an affine basis of an affine space if and only if } is a linear basis of the associated vector space.

FactSnippet No. 1,513,049

Further, transformations of projective space that preserve affine space yield transformations of affine space.

FactSnippet No. 1,513,050