1 Basic definition
An affine subspace is a subset of an affine space that behaves like a linear subspace after a translation. Informally, it is a “shifted” vector subspace: it has the same shape and direction as a linear subspace, but it need not contain the origin. Familiar geometric figures such as ordinary lines, planes, and higher-dimensional flats are affine subspaces when viewed in an affine setting.
The key idea is that affine subspaces are defined by combinations of points rather than by vectors alone. They are exactly the sets that remain stable under affine combinations, which makes them a natural language for geometry that does not depend on a chosen origin.
1.1 Affine spaces and vector spaces
A vector space has a distinguished point, the zero vector, and its elements can be added together and scaled. An affine space, by contrast, is a collection of points with no preferred origin. Differences of points in an affine space form vectors, so vector spaces appear as the “displacement” structure underlying affine geometry.
Because no origin is built into an affine space, a subset can be linear only after one chooses a reference point. Affine subspaces capture the subsets that look linear from any chosen viewpoint, provided one measures them relative to a point on the set.
1.2 Translation of a linear subspace
If \(V\) is a vector space and \(L\) is a linear subspace, then for any vector \(a\), the set \(a + L = \{a + x : x \in L\}\) is an affine subspace. This construction translates the subspace \(L\) by the vector \(a\). The resulting set has the same dimension as \(L\), but it may no longer pass through the origin.
Conversely, every affine subspace can be written in this form once a point on the subspace is selected. This is the standard way to describe affine subspaces in coordinate geometry and linear algebra.
1.3 Closure under affine combinations
A set is an affine subspace if, whenever it contains points \(x_1, \dots, x_n\), it also contains every affine combination \[ \lambda_1 x_1 + \cdots + \lambda_n x_n \] with coefficients satisfying \(\lambda_1 + \cdots + \lambda_n = 1\). This condition says that the set is closed under weighted averages.
This closure property distinguishes affine subspaces from arbitrary translated sets. It reflects the fact that straightness and flatness are preserved when taking points along the same geometric object.
1.4 Equivalent characterizations
Several equivalent descriptions are commonly used. An affine subspace may be defined as a translate of a linear subspace, as a subset closed under affine combinations, or as the solution set of a consistent system of linear equations. Each viewpoint highlights a different aspect of the same concept.
These characterizations are often interchangeable in practice. For example, the translate form is convenient for geometry, while the linear-equation form is useful for computation and algebraic analysis.
2 Examples
Affine subspaces include the most familiar “flat” objects in geometry. In Euclidean settings, they appear as points, lines, planes, and their higher-dimensional analogues. In algebra, they also arise naturally as sets of solutions to linear constraints.
2.1 Affine lines
An affine line is a one-dimensional affine subspace. It can be described as the set of points \(p + tv\), where \(p\) is a point on the line, \(v\) is a nonzero direction vector, and \(t\) ranges over the scalars. If \(p\) is not the origin, the line is not necessarily a linear subspace.
Affine lines are fundamental because they represent the simplest nontrivial example of translation without changing direction. They also serve as building blocks for more complicated affine objects.
2.2 Affine planes
An affine plane is a two-dimensional affine subspace. In three-dimensional space, it is the familiar flat surface that may or may not pass through the origin. It can be specified by a point and two independent direction vectors.
Planes are central in geometry because they capture two-dimensional structure inside higher-dimensional spaces. They are also the geometric form of many linear constraints in applications.
2.3 Hyperplanes
A hyperplane is an affine subspace of codimension one. In an \(n\)-dimensional space, it has dimension \(n-1\). In Euclidean space, hyperplanes generalize the idea of a plane in three dimensions.
Hyperplanes are often described by a single linear equation. They divide space into two half-spaces and play an important role in geometry, optimization, and classification problems.
2.4 Solution sets of linear systems
The set of solutions of a consistent system of linear equations is an affine subspace. If the system is homogeneous, the solution set is a linear subspace; otherwise, it is typically a translated version of one. This is one of the most important algebraic sources of affine subspaces.
Such solution sets may be a point, a line, a plane, or a higher-dimensional flat, depending on the number and independence of the equations. This connection makes affine subspaces a natural framework for linear algebraic computations.
3 Relationship with linear subspaces
Affine subspaces and linear subspaces are closely related, but they differ in whether the origin is part of the structure. A linear subspace must contain the zero vector and is closed under addition and scalar multiplication. An affine subspace need not contain the origin, but it retains a linear direction structure.
3.1 Direction space
Every affine subspace has an associated direction space, which is a linear subspace consisting of all differences of points in the affine set. If \(A\) is an affine subspace and \(p \in A\), then the set \(A - p = \{x - p : x \in A\}\) is a linear subspace. This direction space does not depend on the choice of \(p\).
The direction space determines the orientation of the affine subspace. Two affine subspaces with the same direction space are parallel in the usual geometric sense.
3.2 Choosing an origin
Once a point of an affine space is chosen as an origin, the affine space can be identified with a vector space. Under this identification, an affine subspace becomes a translate of a linear subspace. Different choices of origin lead to different coordinate descriptions, but the underlying affine object remains the same.
This flexibility is useful because many geometric problems become simpler after choosing a convenient reference point. However, the affine viewpoint avoids making the origin artificially special.
3.3 Dimension
The dimension of an affine subspace is defined as the dimension of its direction space. A point has dimension zero, a line has dimension one, a plane has dimension two, and so on. This notion agrees with the familiar geometric intuition.
Dimension is preserved under translation. Therefore, all parallel affine subspaces have the same dimension, even if they are located at different positions in space.
3.4 Parallel affine subspaces
Two affine subspaces are parallel if they have the same direction space. They may coincide or lie in different locations while keeping the same orientation. For example, two distinct horizontal lines in a plane are parallel affine subspaces.
Parallelism is one of the most important relations among affine subspaces. It expresses sameness of direction without requiring overlap.
4 Algebraic properties
Affine subspaces have a useful algebraic structure, especially when combined with the notion of affine combinations. Their behavior under intersection, span, and coordinate expressions makes them suitable for both theoretical and computational work.
4.1 Intersection of affine subspaces
The intersection of two affine subspaces is either empty or an affine subspace. This is consistent with geometric intuition: two lines may meet in a point, coincide, be parallel and disjoint, or intersect in a larger flat if one contains the other. More generally, intersecting affine constraints produces another affine object when the system is compatible.
The dimension of the intersection can be smaller than that of the original subspaces. In many applications, understanding the intersection amounts to solving a linear system.
4.2 Sum and affine span
Given two affine subspaces, one can form the smallest affine subspace containing both, called their affine span. More generally, the affine span of a set is the collection of all affine combinations of its points. It is the affine analogue of linear span.
The affine span of a set of points is the smallest flat that contains them. This construction is frequently used to describe the geometric object determined by a finite collection of points.
4.3 Affine independence
A set of points is affinely independent if no point lies in the affine span of the others. Equivalently, after choosing one point as a reference, the difference vectors to the other points are linearly independent. This notion generalizes linear independence to point configurations.
Affine independence is especially important in geometry because it characterizes the vertices of simplices. In one dimension, two distinct points are affinely independent; in two dimensions, three noncollinear points are affinely independent.
4.4 Barycentric coordinates
Barycentric coordinates express a point as an affine combination of a chosen affinely independent set of points. The coefficients sum to one, and they provide a coordinate system adapted to the geometry of a simplex or affine frame. Such coordinates are often useful in interpolation and geometric computation.
These coordinates are unique when the reference points are affinely independent. They make it easy to describe positions inside triangles, tetrahedra, and higher-dimensional simplices.
5 Coordinates and equations
Affine subspaces can be described in coordinate form, which makes them easy to compute with. Depending on the problem, one may use parametric equations, implicit equations, or a mixture of both.
5.1 Parametric representation
A parametric representation describes an affine subspace as a point plus all linear combinations of a set of direction vectors. For example, \[ x = p + t_1 v_1 + \cdots + t_k v_k \] represents a \(k\)-dimensional affine subspace. The parameters \(t_1, \dots, t_k\) vary over the scalar field.
This form is convenient when one wants to generate points on the subspace. It is also well suited to describing motion along the subspace.
5.2 Implicit representation
An implicit representation describes an affine subspace as the common zero set of one or more affine linear functions. In coordinates, this often appears as a system of linear equations with constants. The resulting set is the intersection of several affine hyperplanes.
Implicit forms are useful for determining whether a point lies in the subspace. They are also effective for computing intersections and analyzing constraints.
5.3 Affine equations in Euclidean space
In Euclidean space, an affine subspace may be defined by equations of the form \[ a_1 x_1 + \cdots + a_n x_n = b. \] A single such equation defines a hyperplane when the coefficient vector is nonzero. Several independent equations define lower-dimensional affine subspaces.
These equations describe flats that are generally displaced from the origin. The constants \(b\) encode the translation relative to the corresponding linear subspace.
5.4 Converting between forms
Parametric and implicit representations can often be converted into one another. A basis for the direction space gives a parametric description, while equations for orthogonality to the direction space give an implicit description. The conversion is a standard task in linear algebra.
Such transformations are important in applications because different forms are more convenient for different operations. Parametric descriptions support point generation, whereas implicit descriptions support constraint checking.
6 Affine maps and transformations
Affine maps are functions that preserve affine combinations. They include translations, linear transformations, and compositions of the two. Affine subspaces are natural objects to study under these maps because their defining properties are preserved.
6.1 Preservation under affine maps
The image of an affine subspace under an affine map is again an affine subspace, possibly of smaller dimension if the map is not injective. This preservation property is one reason affine geometry is so robust. It ensures that flats remain flats under affine transformations.
Because affine maps respect straightness and parallelism, they are the natural symmetries of affine space. Under such maps, lines go to lines, planes to planes, and more general affine subspaces to affine subspaces.
6.2 Images and preimages
The image of an affine subspace under an affine map is obtained by applying the map pointwise. Preimages of affine subspaces are also affine subspaces, provided the map is affine. This makes affine subspaces stable under both forward and inverse geometric constraints.
These facts are frequently used when solving geometric problems by changing coordinates or applying transformations. They allow one to move between equivalent affine configurations.
6.3 Fixed-point sets
The set of fixed points of an affine transformation is often an affine subspace, though it may be empty. If the transformation has fixed points, then the collection of all such points is described by a system of affine equations. In many cases, the fixed-point set is a line, plane, or higher-dimensional flat.
Fixed-point sets are important in the study of symmetries and iterative processes. Their affine nature reflects the linear structure hidden inside the transformation.
6.4 Invariance of dimension and parallelism
Affine transformations preserve dimension of affine subspaces when they are invertible. They also preserve parallelism, since parallel subspaces have the same direction structure, which affine maps carry to equivalent directions. As a result, many geometric properties are invariant under affine change of coordinates.
This invariance is one of the defining features of affine geometry. It allows geometric statements to be made independently of metric notions such as angles or lengths.
7 Applications
Affine subspaces appear throughout mathematics and its applications. They provide a common framework for describing linear constraints, geometric configurations, and feasible regions in optimization problems.
7.1 Linear algebra
In linear algebra, affine subspaces organize the solution sets of systems of equations and the geometry of vector spaces after translation. They help distinguish between homogeneous and nonhomogeneous systems. They also provide a natural setting for studying rank, dimension, and consistency.
Affine reasoning often simplifies calculations by separating the choice of a base point from the linear behavior of directions. This separation is especially valuable when describing families of solutions.
7.2 Analytic geometry
In analytic geometry, affine subspaces are the coordinate expressions of lines, planes, and hyperplanes. They are used to formulate geometric objects by equations rather than by drawings. This approach makes it possible to compute intersections, distances, and relative positions.
Because affine descriptions do not require the origin to be special, they are well adapted to translated shapes. This makes them especially useful in coordinate-based geometry.
7.3 Optimization and feasible regions
In optimization, affine subspaces commonly describe equality constraints. The set of all points satisfying these constraints is often an affine subspace, which may then be intersected with inequalities to form a feasible region. This structure is central in linear programming and related fields.
Affine subspaces can also describe tangent approximations and constraint manifolds in more advanced settings. Their linear structure makes them convenient for algorithm design and analysis.
7.4 Computational geometry
In computational geometry, affine subspaces are used to represent flats, partition space, and solve intersection problems. Algorithms often rely on converting between parametric and implicit forms or on computing affine spans of point sets. Efficient handling of affine subspaces is important in modeling and numerical computation.
They also appear in mesh generation, collision detection, and point-location routines. The stability of affine structure under transformations makes these objects especially practical for computation.