# Chapter 31. Affine Spaces

# Chapter 31. Affine Spaces

An affine space is a geometric space in which differences of points are vectors, but points themselves are not treated as vectors. It has directions, translations, lines, planes, and parallelism, but it has no distinguished zero point unless one is chosen. A vector space can be viewed as an affine space by forgetting the special role of the origin.

Affine spaces are useful because many geometric objects do not naturally pass through the origin. Lines such as

$$
y=2x+1
$$

and solution sets of nonhomogeneous systems such as

$$
Ax=b
$$

are usually not vector subspaces. They are affine subspaces.

## 31.1 Points and Vectors

In a vector space, the elements are vectors. They can be added and multiplied by scalars.

In an affine space, the elements are points. Points cannot be added in the same direct way. Instead, one may subtract two points to obtain a vector, or add a vector to a point to obtain another point.

If \(P\) and \(Q\) are points, then the displacement from \(P\) to \(Q\) is written

$$
\overrightarrow{PQ}.
$$

This is a vector.

If \(P\) is a point and \(v\) is a vector, then

$$
P+v
$$

is the point obtained by translating \(P\) by \(v\).

Thus affine geometry separates two ideas that are often conflated in \(\mathbb{R}^n\):

| Object | Meaning |
|---|---|
| Point | A location |
| Vector | A displacement |
| \(Q-P\) or \(\overrightarrow{PQ}\) | Vector from \(P\) to \(Q\) |
| \(P+v\) | Point reached from \(P\) by displacement \(v\) |

## 31.2 Definition

Let \(V\) be a vector space over a field \(F\). An affine space modeled on \(V\) is a set \(A\), whose elements are called points, together with an operation

$$
A\times V\to A,
\qquad
(P,v)\mapsto P+v,
$$

satisfying the following conditions.

First,

$$
P+0=P
$$

for every point \(P\in A\).

Second,

$$
(P+v)+w=P+(v+w)
$$

for all \(P\in A\) and all \(v,w\in V\).

Third, for every pair of points \(P,Q\in A\), there exists a unique vector \(v\in V\) such that

$$
P+v=Q.
$$

This unique vector is denoted

$$
\overrightarrow{PQ}.
$$

Equivalently,

$$
Q=P+\overrightarrow{PQ}.
$$

This definition says that \(V\) acts on \(A\) by translations, and that any point can be moved to any other point by one unique translation.

## 31.3 Basic Identities

The displacement vectors in an affine space satisfy familiar rules.

For any points \(P,Q,R\in A\),

$$
\overrightarrow{PP}=0,
$$

$$
\overrightarrow{PQ}+\overrightarrow{QR}=\overrightarrow{PR},
$$

and

$$
\overrightarrow{QP}=-\overrightarrow{PQ}.
$$

The second identity is the head-to-tail rule for displacements. Moving from \(P\) to \(Q\), then from \(Q\) to \(R\), gives the same total displacement as moving directly from \(P\) to \(R\).

These identities follow from the translation axioms. They are the algebraic rules behind elementary geometry.

## 31.4 Vector Spaces as Affine Spaces

Every vector space \(V\) can be treated as an affine space modeled on itself.

The points are the elements of \(V\). The translation operation is ordinary vector addition:

$$
P+v
$$

where \(P\in V\) is now regarded as a point and \(v\in V\) as a displacement vector.

The difference between two points is

$$
\overrightarrow{PQ}=Q-P.
$$

This is the usual subtraction of vectors.

However, when \(V\) is viewed as an affine space, the zero vector has no intrinsic geometric privilege. It is merely one possible point chosen as origin.

This viewpoint is useful because geometry often should not depend on a chosen origin.

## 31.5 Choosing an Origin

An affine space becomes identified with its model vector space once an origin point is chosen.

Let \(O\in A\) be fixed. Define

$$
\Phi_O:A\to V
$$

by

$$
\Phi_O(P)=\overrightarrow{OP}.
$$

This assigns to each point \(P\) its coordinate vector relative to the chosen origin \(O\).

The map \(\Phi_O\) is a bijection. Every point corresponds to a unique displacement from \(O\).

But the identification depends on \(O\). Choosing a different origin changes all coordinate vectors by a translation.

Thus an affine space has no canonical zero point. A coordinate origin is extra structure.

## 31.6 Affine Subspaces

An affine subspace is a translate of a linear subspace.

Let \(V\) be a vector space, let \(U\subseteq V\) be a linear subspace, and let \(p\in V\). The set

$$
p+U=\{p+u:u\in U\}
$$

is an affine subspace.

If \(p\in U\), then

$$
p+U=U,
$$

so the affine subspace is actually a linear subspace.

If \(p\notin U\), then \(p+U\) does not contain the origin and is not a vector subspace.

Affine subspaces are the natural setting for lines and planes not necessarily passing through the origin.

## 31.7 Lines

An affine line has the form

$$
P+\operatorname{span}(v),
$$

where \(P\) is a point and \(v\neq 0\) is a direction vector.

Equivalently,

$$
L=\{P+tv:t\in F\}.
$$

In \(\mathbb{R}^2\), if

$$
P=
\begin{bmatrix}
1\\
2
\end{bmatrix},
\qquad
v=
\begin{bmatrix}
3\\
1
\end{bmatrix},
$$

then the affine line is

$$
L=
\left\{
\begin{bmatrix}
1\\
2
\end{bmatrix}
+
t
\begin{bmatrix}
3\\
1
\end{bmatrix}
:t\in\mathbb{R}
\right\}.
$$

This line generally does not pass through the origin. It is a translate of the one-dimensional subspace \(\operatorname{span}(v)\).

## 31.8 Planes

An affine plane in \(\mathbb{R}^3\) has the form

$$
P+\operatorname{span}(u,v),
$$

where \(u\) and \(v\) are linearly independent direction vectors.

Thus

$$
\Pi=\{P+su+tv:s,t\in\mathbb{R}\}.
$$

For example,

$$
P=
\begin{bmatrix}
1\\
0\\
2
\end{bmatrix},
\qquad
u=
\begin{bmatrix}
1\\
1\\
0
\end{bmatrix},
\qquad
v=
\begin{bmatrix}
0\\
1\\
1
\end{bmatrix}.
$$

Then

$$
\Pi=
\left\{
\begin{bmatrix}
1\\
0\\
2
\end{bmatrix}
+
s
\begin{bmatrix}
1\\
1\\
0
\end{bmatrix}
+
t
\begin{bmatrix}
0\\
1\\
1
\end{bmatrix}
:s,t\in\mathbb{R}
\right\}.
$$

The direction subspace of this plane is

$$
\operatorname{span}(u,v).
$$

## 31.9 Direction Subspace

Every affine subspace has an associated direction subspace.

If

$$
S=p+U,
$$

where \(U\) is a linear subspace, then \(U\) is called the direction space of \(S\).

The direction space can be recovered from \(S\) as the set of all differences of points in \(S\):

$$
U=\{\overrightarrow{PQ}:P,Q\in S\}.
$$

Thus an affine subspace consists of one base point plus all directions in a linear subspace.

The base point is not unique. If \(q\in p+U\), then

$$
p+U=q+U.
$$

The direction subspace is unique.

## 31.10 Affine Combinations

An affine combination of points \(P_1,\ldots,P_k\) is an expression

$$
\lambda_1P_1+\cdots+\lambda_kP_k
$$

where the scalars satisfy

$$
\lambda_1+\cdots+\lambda_k=1.
$$

This condition makes the expression independent of the choice of origin.

For two points \(P\) and \(Q\), the affine combinations are

$$
(1-t)P+tQ.
$$

As \(t\) varies over \(\mathbb{R}\), these points form the affine line through \(P\) and \(Q\).

When

$$
0\leq t\leq 1,
$$

the points form the line segment from \(P\) to \(Q\).

## 31.11 Why Coefficients Sum to One

The condition

$$
\lambda_1+\cdots+\lambda_k=1
$$

is what makes affine combinations compatible with translations.

If every point is shifted by the same vector \(a\), then

$$
\lambda_1(P_1+a)+\cdots+\lambda_k(P_k+a)
$$

equals

$$
\lambda_1P_1+\cdots+\lambda_kP_k
+
(\lambda_1+\cdots+\lambda_k)a.
$$

If the coefficients sum to \(1\), this becomes

$$
\lambda_1P_1+\cdots+\lambda_kP_k+a.
$$

So the affine combination shifts by the same amount as the original points. It does not depend on a hidden choice of origin.

## 31.12 Convex Combinations

A convex combination is an affine combination whose coefficients are nonnegative:

$$
\lambda_i\geq 0,
\qquad
\lambda_1+\cdots+\lambda_k=1.
$$

Convex combinations describe points inside the convex hull of a set.

For two points,

$$
(1-t)P+tQ,
\qquad
0\leq t\leq 1,
$$

is the line segment between \(P\) and \(Q\).

For three noncollinear points in the plane, convex combinations describe the filled triangle with those vertices.

Affine combinations describe flats such as lines and planes. Convex combinations describe bounded geometric regions inside them.

## 31.13 Affine Span

The affine span of points \(P_1,\ldots,P_k\) is the set of all affine combinations of those points:

$$
\operatorname{aff}(P_1,\ldots,P_k) =
\left\{
\lambda_1P_1+\cdots+\lambda_kP_k:
\lambda_1+\cdots+\lambda_k=1
\right\}.
$$

It is the smallest affine subspace containing the points.

For two distinct points, the affine span is the line through them.

For three noncollinear points in \(\mathbb{R}^3\), the affine span is the plane through them.

For points \(P_0,P_1,\ldots,P_k\), the affine span can also be written as

$$
P_0+\operatorname{span}
(\overrightarrow{P_0P_1},\ldots,\overrightarrow{P_0P_k}).
$$

This formula reduces affine span to ordinary linear span of displacement vectors.

## 31.14 Affine Independence

Points

$$
P_0,P_1,\ldots,P_k
$$

are affinely independent if the displacement vectors

$$
\overrightarrow{P_0P_1},\ldots,\overrightarrow{P_0P_k}
$$

are linearly independent.

Equivalently, no point lies in the affine span of the others.

Examples:

| Points | Affine independence condition |
|---|---|
| Two points | Distinct |
| Three points | Not collinear |
| Four points in \(\mathbb{R}^3\) | Not coplanar |

Affine independence is the point-based analogue of linear independence.

## 31.15 Dimension of an Affine Subspace

The dimension of an affine subspace

$$
S=p+U
$$

is defined as

$$
\dim S=\dim U.
$$

Thus:

| Affine subspace | Dimension |
|---|---:|
| A point | 0 |
| A line | 1 |
| A plane | 2 |
| A translate of a \(k\)-dimensional subspace | \(k\) |

The dimension is determined by the direction subspace, not by the chosen base point.

## 31.16 Nonhomogeneous Linear Systems

The solution set of a homogeneous system

$$
Ax=0
$$

is a linear subspace.

The solution set of a nonhomogeneous system

$$
Ax=b
$$

is generally an affine subspace, provided it is nonempty.

If \(x_p\) is one particular solution of \(Ax=b\), then every solution has the form

$$
x=x_p+z,
$$

where

$$
z\in \operatorname{Null}(A).
$$

Therefore

$$
\{x:Ax=b\}=x_p+\operatorname{Null}(A).
$$

The direction space of the affine solution set is the null space of \(A\).

## 31.17 Example: An Affine Solution Line

Consider

$$
x+2y=5.
$$

One particular solution is

$$
p=
\begin{bmatrix}
5\\
0
\end{bmatrix}.
$$

The associated homogeneous equation is

$$
x+2y=0.
$$

Its solution space is

$$
\operatorname{span}
\left(
\begin{bmatrix}
-2\\
1
\end{bmatrix}
\right).
$$

Therefore the full solution set is

$$
\begin{bmatrix}
5\\
0
\end{bmatrix}
+
\operatorname{span}
\left(
\begin{bmatrix}
-2\\
1
\end{bmatrix}
\right).
$$

Equivalently,

$$
\left\{
\begin{bmatrix}
5\\
0
\end{bmatrix}
+
t
\begin{bmatrix}
-2\\
1
\end{bmatrix}
:t\in\mathbb{R}
\right\}.
$$

This is a line not passing through the origin, so it is an affine subspace but not a linear subspace.

## 31.18 Affine Maps

An affine map is a function that preserves affine combinations.

Between vector spaces, an affine map has the form

$$
T(x)=Ax+b,
$$

where \(A\) is linear and \(b\) is a fixed vector.

If \(b=0\), then \(T\) is linear. If \(b\neq 0\), then \(T\) includes a translation.

For example,

$$
T(x,y)=
\begin{bmatrix}
2&0\\
0&3
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}
+
\begin{bmatrix}
1\\
-4
\end{bmatrix}
$$

first scales the coordinate directions and then translates the result.

Affine maps send affine subspaces to affine subspaces.

## 31.19 Linear Versus Affine

The difference between linear and affine is the treatment of the origin.

A linear map satisfies

$$
L(0)=0.
$$

An affine map need not. If

$$
T(x)=Ax+b,
$$

then

$$
T(0)=b.
$$

Thus affine maps preserve lines and parallelism, but they may move the origin.

| Concept | Linear | Affine |
|---|---|---|
| Basic objects | Vectors | Points |
| Distinguished origin | Yes | No |
| Typical map | \(x\mapsto Ax\) | \(x\mapsto Ax+b\) |
| Subspaces | Pass through origin | May be translated |
| Combinations | Coefficients arbitrary | Coefficients sum to \(1\) |
| Solution sets | \(Ax=0\) | \(Ax=b\) |

## 31.20 Summary

An affine space is a space of points modeled on a vector space of displacements. It supports translation and difference of points, but it has no distinguished zero point until one is chosen.

The key ideas are:

| Concept | Meaning |
|---|---|
| Affine space | Point space modeled on a vector space |
| Model vector space | Space of displacement vectors |
| \(\overrightarrow{PQ}\) | Vector from point \(P\) to point \(Q\) |
| \(P+v\) | Translation of point \(P\) by vector \(v\) |
| Affine subspace | Translate \(p+U\) of a linear subspace |
| Direction space | The linear subspace \(U\) in \(p+U\) |
| Affine combination | Combination with coefficients summing to \(1\) |
| Affine span | Smallest affine subspace containing given points |
| Affine independence | Linear independence of displacement vectors |
| Affine map | Map of the form \(x\mapsto Ax+b\) |

Affine spaces are the natural language for geometry without a preferred origin. They explain translated lines and planes, nonhomogeneous solution sets, affine maps, barycentric coordinates, and the distinction between locations and displacements.