# Affine Space

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An Affine Space is a vector space/metric space without an origin point.

## References

### 2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/affine_space Retrieved:2015-6-9.
• In mathematics, an affine space is a geometric structure that generalizes certain properties of parallel lines in Euclidean space. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors between two points of the space. Thus it makes sense to subtract two points of the space, giving a vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a vector to a point of an affine space, resulting in a new point displaced from the starting point by that vector.

The simplest example of an affine space is a linear subspace of a vector space that has been translated away from the origin. In finite dimensions, such an affine subspace corresponds to the solution set of an inhomogeneous linear system. The displacement vectors for that affine space live in the solution set of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space.

### 2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Affine_space Retrieved:2014-9-27.
• In mathematics, an affine space is a geometric structure that generalizes certain properties of parallel lines in Euclidean space. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors between two points of the space. Thus it makes sense to subtract two points of the space, giving a vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a vector to a point of an affine space, resulting in a new point displaced from the starting point by that vector.

The simplest example of an affine space is a linear subspace of a vector space that has been translated away from the origin. In finite dimensions, such an affine subspace corresponds to the solution set of an inhomogeneous linear system. The displacement vectors for that affine space live in the solution set of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space.

### 2009

• Weisstein, Eric W. “Affine Space.” From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/AffineSpace.html
• Let V be a vector space over a field K, and let A be a nonempty set. Now define addition p+a in A for any vector a in V and element p in A subject to the conditions:
• 1. p+0=p.
• 2. (p+a)+b=p+(a+b).
• 3. For any q in A, there exists a unique vector a in V such that q=p+a.
• Here, a, b in V. Note that (1) is implied by (2) and (3). Then A is an affine space and K is called the coefficient field.
• 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 n-tuple of its coordinates. Every ordered pair of points A and B in an affine space is then associated with a vector AB.