Affine Spaces

UNDER CONSTRUCTION

Recall that a vector space consists of a set of objects \( V \) called vectors, a field \( \mathbb{F} \) and two operations on the vectors, addition and scalar multiplication. The vectors form an additive group under addition and scalar multiplication is a binary operation \( \mathbb{F} \times V \to V \) which distributes over addition. In a vector space, if coordinates are used and a norm is assigned, then every vector therein is uniquely defined by its norm and its direction, whence these two properties are invariant under translation of the vectors. It follows that in any vector space there is a single origin, namely \( \vec{0} \). In an affine space however, this zero origin is no longer necessarily the single origin, and translation of the elements of an affine space, called points, will result in different points, whence the idea of multiple origins originates.

Definition. An affine space is a triple \( (A, V, +) \) where \( A \) is a set of objects called points and \( V \) is a vector space with the following properties:

1. \( \forall a \in A, \vec{v}, \vec{w} \in V, a + ( \vec{v} + \vec{w} ) = (a + \vec{v}) + \vec{w} \)

2. \( \forall a \in A, \vec{0} \in V, a + \vec{0} = a \)

3. \( \forall a, b \in A, \exists \vec{v} \in V \) such that \( a = b + \vec{v} \).

It is apparent that the additive group \( V \) induces a transitive group action upon \( A \); this directly follows from the definition of a group action. Condition 3 tells us that the difference of two points \( a - b \) results in a vector \( \vec{v} \), and this is where the idea of multiple origins arises from in an affine space.

Definition. Let \( (A_1, V_1, +), (A_2, V_2, +) \) be affine spaces. \( f: A_1 \to A_2 \) is an affine transformation if \( \forall a_1, a_2 \in A_1, f(a_1) - f(a_2) = Df(a_1 - a_2) \), where \( Df: V_1 \to V_2 \) is a linear transformation. \( Df \) is called the linear part of the affine transformation.

Example. The transformation \( f_{\vec{v}} : A \to A , a \mapsto a + \vec{v} \) is affine. It is obvious.

Proposition 1. The collection of affine spaces form a category.

Proof. This follows from definition. \( \square \)

Proposition 2. The category of affine spaces and of vector spaces are functorial.

Proof. Let \( (A_i, V_i, + )_{i=1, 2}\) be two affine spaces, wherebetween \( f: A_1 \to A_2 \) is affine and \( Df: V_1 \to V_2 \) linear. A mapping associating \( f \) with \( Df \) is a functor from the category of affine spaces into that of vector spaces. This follows from proposition 1 and the obvious fact that the composition of two affine transformations are affine, whence the proposition is proved. \( \square \)

Proposition 3. There exists a unique affine transformation \( f \) between \( (A_1, V_1, +), (A_2, V_2, +) \) determined by two points \( a_i \in A_{i = 1, 2} \) and a linear map \( T: V_1 \to V_2 \) with \( f(a_1) = a_2 \).

Proof. By 3) in the definition of affine space, every element \( a \in A_1 \) may be written \( a = a_1 + \vec{v} \). Let \( f \) have the following property: \( f(a ) - a_2 = f(a_1 + \vec{v}) - a_2 = T(\vec{v}) \). Then \( f \) is obviously affine, \( Df = T\), and \( f(a_1) = a_2 \). Uniqueness is obvious. The proposition is proved. \( \square \).

Proposition 4. Let \( (A_i, V_i, +)_{i=1, 2} \) be two affine spaces. Two affine transformations \( f_1, f_2 : A_1 \to A_2 \) have the same linear part if and only if \( f_1 = f_{\vec{v}} \circ f_2 \) for some unique \( \vec{v} \in V_2 \).

Proof. The reverse direction follows from definition. To show the forwards direction, set \( f'_2 = f_{\vec{f_2(a) - f_1(a)}} \circ f_1 \) and by proposition 3, \( f'_2 = f_2 \). The proposition is proved. \( \square \)

Consider the affine spaces \( (A, V, +), (V, V, +) \), and affine transformation \( f: A \to V, Df = \text{id}_V \).

Kostrikin, A., and Manin, Y. "Linear Algebra and Geometry".