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An affine space is a concept in linear algebra: a set of points which, if displaced so that it contained the origin, would be a subspace of the vector space in which it lies. All hyperplanes which do not intersect the origin are affine spaces.

All affine spaces are of the form b + Sum over basis vectors {an xn} with a's scalar coefficients, the x's forming a basis for the subspace the affine space is parallel to, and b being any chosen point in the affine space.

Each affine space has one point B which is closest to the origin. The dot product of any vector in the affine space with B is a constant. As such, an affine space can be considered as a set of points X in the vector space under a set of constraints of the form X•A = n where A is a vector, and n is some scalar. It is always possible to choose the set of constraints so that only one has nonzero n. In that case, the A for that constraint equals the vector B mentioned earlier.

Note that affine spaces have different properties than standard subspaces. For example, adding two vectors in an affine space using conventional addition produces a result which is not in the affine space (you have replaced B with 2B).

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