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.
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The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace.
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Dimension of an affine space is defined as the dimension of the vector space of its translations.
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An affine space of dimension one is an affine line.
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An affine subspace of dimension in an affine space or a vector space of dimension is an affine hyperplane.
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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.
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The vector space is said to be associated to the affine space, and its elements are called vectors, translations, or sometimes free vectors.
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An affine transformation or endomorphism of an affine space is an affine map from that space to itself.
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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 .
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An or barycentric frame of an affine space is a generating set that is independent .
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Equivalently, } is an affine basis of an affine space if and only if } is a linear basis of the associated vector space.
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Further, transformations of projective space that preserve affine space yield transformations of affine space.
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