3. Affine spaces

Cosets of a linear subspace

If W is a linear subspace of a vector space V over K and v\in V, then the set

    \[v+W=\{v+w\colon w\in W\}\]

is called a coset of W.

Eg. (1,1)+lin(2,-1)=\{(1+2t,1-t)\colon t\in \mathebb{R}\} is a coset of the line lin(2,-1) on the plave.

You can also see the cosets as equivalence classes of relation v\sim v' iff v-v'\in W.

Notice that a system of equations describing a coset of a subspace W has the same left-hand side as the system describing W, but it is not necessary homogenous — the right-hand side can be chosen in such a way that a given vector satisfies it and thus is in the coset..

Affine combinations

Given points p_0,\ldots p_n\in V, we call a_0p_0+\ldots +a_np_n their affine combination, if a_0+\ldots +a_n=1.

It is easy to notice that a subset of a vector space is closed under taking affine combination if and only if it is a coset of a linear subspace. Notice also that an affine combination of affine combinations is an affine combination.

The set of all affine combinations of points p_0,\ldots, p_n is denoted by af(p_0,\ldots, p_n).

Points p_0,\ldots, p_n are affinally independent, if none of them is an affine combination of others.

Affine subspaces of a linear space

Till now we have dealt with linear spaces, in particular in all our lines, planes and other subspaces the zero vector was always included. But obviously it makes sense to study also such spaces but translated from zero by a given vector. Such subspaces (are no longer linear) are called affine subspaces of a linear space. The term hyperplane is used to describe affine subspaces of dimension one less then the whole linear space we work in.

So if V is a linear subspace and v is a vector, then M=v+V (meaning subspace V translated by v), is an affine subspace. The space V will be called the tangent space (or the direction) to M and denoted by T(M) (or \vec{M}).

An affine subspace of a linear space can be described in the following ways:

  • giving a vector and its direction, e.g.: (1,0,-1)+lin((1,2,0),(0,1,1)).
  • giving a point basis (points it goes through) (we need one point more then the dimension of considered subspace and those n+1 points have to be affinally independent), e.g.: (1,0,-1),(2,2,-1),(1,1,0) (sometimes denoted by af((1,0,-1),(2,2,-1),(1,1,0))).
  • giving a parametrization, e.g..: \{(1+t,2t+s,-1+t)\colon t,s\in\mathbb{R}\}
  • giving a system of linear equations, e.g.:

        \[2x-y+z=0\]

As usually we would like to be able to change the form of description.

A vector with a the tangent space and a set of points

Given a translation vector and the tangent space, we can easily get the points, which are included in the considered affine subspace. E.g. if H=(1,0,0,-1)+lin((1,-1,0,1),(2,-1,1,0)), it can be also defined as the only affine subspace including points: (1,0,0,-1),(1,0,0,-1)+(1,-1,0,1)=(2,-1,0,0), (1,0,0,-1)+(2,-1,1,0)=(3,-1,1,-1).

Reversely, if H goes through: (1,0,0,-1),(2,-1,0,0),(3,-1,1,-1), choose one of those points as a translation vector, and we get that H=(1,0,0,-1)+lin((2,-1,0,0)-(1,0,0,-1),(3,-1,1,-1)-(1,0,0,-1))=(1,0,0,-1)+lin((1,-1,0,1),(2,-1,1,0)).

A translation vector with the direction and parametrization

Given a translation vector and vectors spanning the tangent space we can easily get a parametrization. E.g. if H=(1,0,0,-1)+lin((1,-1,0,1),(2,-1,1,0)), then every vector of H is of form (1,0,0,-1)+t(1,-1,0,1)+s(2,-1,1,0), so H=\{(1+t+2s,-t-s,s,-1+t)\colon s,t\in\mathbb{R}\}.

Reversely, given H=\{(1+t+2s,-t-s,s,-1+t)\colon s,t\in\mathbb{R}\} we know that all the vectors of H are of form (1,0,0,-1)+t(1,-1,0,1)+s(2,-1,1,0), so H=(1,0,0,-1)+lin((1,-1,0,1),(2,-1,1,0)).

From a translation vector and the direction to a system of linear equations

The system of linear equations describing a given affine subspace differ from the system of equation describing its tangent space only by the column of free coefficients (because the difference between two points from H always is in T(H), so the difference of two solutions of the system of linear equations we are looking for is a solution of the system of linear equations describing the tangent space) — and in the second space we have an uniform system which we already know how to find. Therefore start with finding the system describing T(H). For example, let H=(1,0,0,-1)+lin((1,-1,0,1),(2,-1,1,0)). First find the system describing T(H)=lin((1,-1,0,1),(2,-1,1,0)) — we already know how to find such a system and so T(H) is given by:

    \[\begin{cases}-x-y+z=0\\x+2y+w=0\end{cases}\]

The given translation vector has to be a solution of the system we are looking for, so we have to choose the free coefficients in such a way that it will be true. So we can simply substitute the translation vector into the left sides of the equations: -1-0+0=-1, 1+0+-1=0. Therefore, the system of equations we are looking for is the following:

    \[\begin{cases}-x-y+z=-1\\x+2y+w=0\end{cases}\]

From a system of equations to a parametrization

Notice, that simply given a system of equations for H, its general solution is a parametrization of H!

Affine subspace

A subset G of an affine space H is its affine subspace, if it is closed under taking affine combinations of (finitely many) points.

Point basis and basic system

We say that a system of points p_0,\ldots, p_n spans an affine subspace H, if H=af(p_0,\ldots, p_n).

An affinally independent system p_0,\ldots, p_n, which spans an affine subspace H is called its point basis. Then for every point x\in H there are unique a_0,\ldots, a_n such that x=a_0p_0+\ldots+ a_np_n and a_0+\ldots+a_n=1. These are the coordinates of x in this point basis.

For points p,p', the vector p'-p is sometimes denoted by \overrightarrow{pp'}.

Consider a coset of a linear subspace v+L. Then, if v_1,\ldots, v_n is a basis of L, the system v;v_1,\ldots, v_n is called the basic system of this coset. Notice that then v, v+v_1,\ldots, v+v_n is its point basis. And also, if p_0, \ldots, p_n is its point basis, then p_0;\overrightarrow{p_0p_1},\ldots, \overrightarrow{p_0p_n} is its basic system.

If v;v_1,\ldots, v_n is a basic system of a coset H and x\in H i x=v+a_1v_1+\ldots +a_nv_n, then a_1,\ldots , a_n are unique and are called the coordinates of x with respect to this basic system.