Affine space/Introduction/Section

Linear subspaces of a vector space always contain ${}0$. Hence, a line ${}G\subset \mathbb {R} ^{2}$ not running through the zero point, is not a linear subspace. However, it is still a "linear object“, which should be studied in the framework of linear algebra.

Definition

Let ${}V$ be a vector space. An affine subspace of ${}V$ is (the empty set or) a subset of the form

${}P+U={\left\{P+u\mid u\in U\right\}}\,,$

where ${}U\subseteq V$ is a linear subspace and ${}P\in V$

is a vector.

The point ${}P$ is called a starting point, and the linear subspace ${}U$ is called the translation space, or simply the underlying linear subspace. The points in an affine space should be thought of as locations, the points of ${}U$ should be thought of as translation vectors. One might discuss whether the empty set should be allowed as an affine (sub)space, the following remark, the definition and fact speak in favor of allowing this.

Remark

The solution set of a system of inhomogeneous linear equations in ${}n$ variables is an affine subspace of ${}K^{n}$. The underlying vector space in the solution space of the corresponding homogeneous system.

Example

For a linear mapping

$\varphi \colon V\longrightarrow W$

between ${}K$-vector spaces ${}V$ and ${}W$ and an element ${}Q\in W$, the preimage for ${}Q$ (the fiber over ${}Q$)

${}\varphi ^{-1}(Q)={\left\{P\in V\mid \varphi (P)=Q\right\}}\,$

is an affine subspace of ${}V$. If this is non-empty, then we can take any point ${}P_{0}\in V$ with

${}\varphi (P_{0})=Q\,$

as starting point The translation space is the kernel of ${}\varphi$. When a linear mapping is given, then ${}V$ is partitioned in a layered family of parallel affine subspaces.

The action of a translation in the plane on a subset.

A further reasoning yields a somewhat more abstract concept. The natural space can be given coordinates and then it is identified with ${}\mathbb {R} ^{3}$. For this, we have to choose arbitrarily a point in the space as ${}0$. The natural space does not contain a natural zero, and also no natural addition of points. Despite of this, the natural space is tightly related to a vector space, namely the vector space of all translations of the space. Such a translation is an elementary geometric construction, every point of the space is translated by a certain translating vector. A translation is determined by every point together with its image point. The set of all these translations form a vector space, where the addition is given by composing translations. The zero translation is the identity. If a point ${}P$ of the space is fixed, then we get a bijection between the space and the vector space of translations, by attaching a translation vector at ${}P$ and determining the resulting point. Such a fixation is also called a choice of an origin.

Definition

An affine space over a ${}K$-vector space ${}V$ is a set ${}E$, together with a mapping

$V\times E\longrightarrow E,(v,P)\longmapsto P+v,$

which satisfies the following three conditions:

1. ${}P+0=P$ for all ${}P\in E$,
2. ${}(P+v)+w=P+(v+w)$ for all ${}v,w\in V$ and ${}P\in E$,
3. For two points ${}P,Q\in E$, there exists exactly one vector ${}v\in V$ such that ${}Q=P+v$.

This addition is called affine addition or Translation. For two given points ${}P,Q\in E$, the uniquely determined translation vector is denoted by ${}{\overrightarrow {PQ}}$. Beside ${}P+{\overrightarrow {PQ}}=Q$, the following rules holds

1. ${}{\overrightarrow {PP}}=0$ for ${}P\in E$.
2. ${}{\overrightarrow {PQ}}=-{\overrightarrow {QP}}$ for ${}P,Q\in E$.
3. ${}{\overrightarrow {PQ}}+{\overrightarrow {QR}}={\overrightarrow {PR}}$ for ${}P,Q,R\in E$,

where these identities live in the vector space ${}V$, see exercise.

The mapping

$+\colon V\times E\longrightarrow E$

exhibits several aspects. For every point ${}P\in E$, the mapping

$V\longrightarrow E,v\longmapsto P+v,$

is a bijection between the underlying vector space and the affine space. This bijection is not canonical, as it depends on the chosen point. Every vector ${}v\in V$ defines the mapping

$E\longrightarrow E,P\longmapsto P+v,$

which is called the translation on ${}E$ for the vector ${}v$. The mapping

$E\times E\longrightarrow V,(P,Q)\longmapsto {\overrightarrow {PQ}},$

assigns to a pair of points their (uniquely determined) translating vector. Instead of ${}{\overrightarrow {PQ}}$, we sometimes write ${}Q-P$.

Every vector space ${}V$ is also an affine space over itself, with the vector space addition as addition. An affine subspace ${}P+U\subseteq V$ in the sense of Definition is an affine space over ${}U$.

Example

The solution space of the homogeneous linear equation

${}7x-3y+4z=0\,$

is

${}U=\langle {\begin{pmatrix}3\\7\\0\end{pmatrix}},\,{\begin{pmatrix}0\\4\\3\end{pmatrix}}\rangle \,,$

the solution set of the inhomogeneous linear equation

${}7x-3y+4z=2\,$

is

${}E={\begin{pmatrix}0\\0\\{\frac {1}{2}}\end{pmatrix}}+U={\begin{pmatrix}0\\0\\{\frac {1}{2}}\end{pmatrix}}+\langle {\begin{pmatrix}3\\7\\0\end{pmatrix}},\,{\begin{pmatrix}0\\4\\3\end{pmatrix}}\rangle \,.$

The affine addition is the mapping

$U\times E\longrightarrow E,(u,P)\longmapsto (u+P),$

which assigns to a pair consisting in a solution of the homogeneous equation and a solution of the inhomogeneous equation their sum, which is a solution of the inhomogeneous equation. For two solutions of the inhomogeneous equation, their difference is a solution of the homogeneous equation. For example, for

${}u=-2{\begin{pmatrix}3\\7\\0\end{pmatrix}}+3{\begin{pmatrix}0\\4\\3\end{pmatrix}}={\begin{pmatrix}-6\\-2\\9\end{pmatrix}}\in U\,,$

the point

${}{\begin{pmatrix}0\\0\\{\frac {1}{2}}\end{pmatrix}}+{\begin{pmatrix}-6\\-2\\9\end{pmatrix}}={\begin{pmatrix}-6\\2\\{\frac {19}{2}}\end{pmatrix}}\,$

is another solution in ${}E$. The two solutions ${}{\begin{pmatrix}1\\3\\1\end{pmatrix}}$ and ${}{\begin{pmatrix}2\\4\\0\end{pmatrix}}$ from ${}E$ are related by the translating vector

${}{\begin{pmatrix}2\\4\\0\end{pmatrix}}-{\begin{pmatrix}1\\3\\1\end{pmatrix}}={\begin{pmatrix}1\\1\\-1\end{pmatrix}}\,.$