Affine Subspaces Review¶
For a detailed introduction to affine concepts, see [KW79]. For a vector \(v \in \RR^n\), the function \(f\) defined by \(f (x) = x + v, x \in \RR^n\) is a translation of \(\RR^n\) by \(v\). The image of any set \(\mathcal{S}\) under \(f\) is the \(v\)-translate of \(\mathcal{S}\). A translation of space is a one to one isometry of \(\RR^n\) onto \(\RR^n\).
A translate of a \(d\)-dimensional, linear subspace of \(\RR^n\) is a \(d\)-dimensional flat or simply \(d\)-flat in \(\RR^n\). Flats of dimension 1, 2, and \(n-1\) are also called lines, planes, and hyperplanes, respectively. Flats are also known as affine subspaces.
Every \(d\)-flat in \(\RR^n\) is congruent to the Euclidean space \(\RR^d\). Flats are closed sets.
An affine combination of the vectors \(v_1, \dots, v_m\) is a linear combination in which the sum of coefficients is 1. Thus, \(b\) is an affine combination of \(v_1, \dots, v_m\) if \(b = k_1 v_1 + \dots k_m v_m\) and \(k_1 + \dots + k_m = 1\). The set of affine combinations of a set of vectors \(\{ v_1, \dots, v_m \}\) is their affine span. A finite set of vectors \(\{v_1, \dots, v_m\}\) is called affine independent if the only zero-sum linear combination of theirs representing the null vector is the null combination. i.e. \(k_1 v_1 + \dots + k_m v_m = 0\) and \(k_1 + \dots + k_m = 0\) implies \(k_1 = \dots = k_m = 0\). Otherwise, the set is affinely dependent. A finite set of two or more vectors is affine independent if and only if none of them is an affine combination of the others.
Vectors vs. Points An n-tuple \((x_1, \dots, x_n)\) is used to refer to a point \(X\) in \(\RR^n\) as well as to a vector from origin \(O\) to \(X\) in \(\RR^n\). In basic linear algebra, the terms vector and point are used interchangeably. While discussing geometrical concepts (affine or convex sets etc.), it is useful to distinguish between vectors and points. When the terms “dependent” and “independent” are used without qualification to points, they refer to affine dependence/independence. When used for vectors, they mean linear dependence/independence.
The span of \(k+1\) independent points is a \(k\)-flat and is the unique \(k\)-flat that contains all \(k+1\) points. Every \(k\)-flat contains \(k+1\) (affine) independent points. Each set of \(k+1\) independent points in the \(k\)-flat forms an affine basis for the flat. Each point of a \(k\)-flat is represented by one and only one affine combination of a given affine basis for the flat. The coefficients of the affine combination of a point are the affine coordinates of the point in the given affine basis of the \(k\)-flat. A \(d\)-flat is contained in a linear subspace of dimension \(d+1\). This can be easily obtained by choosing an affine basis for the flat and constructing its linear span.
A function \(f\) defined on a vector space \(V\) is an affine function or affine transformation or affine mapping if it maps every affine combination of vectors \(u, v\) in \(V\) onto the same affine combination of their images. If \(f\) is real valued, then \(f\) is an affine functional. A property which is invariant under an affine mapping is called affine invariant. The image of a flat under an affine function is a flat.
Every affine function differs from a linear function by a translation. A functional is an affine functional if and only if there exists a unique vector \(a \in \RR^n\) and a unique real number \(k\) such that \(f(x) = \langle a, x \rangle + k\). Affine functionals are continuous. If \(a \neq 0\), then the linear functional \(f(x) = \langle a, x \rangle\) and the affine functional \(g(x) = \langle a, x \rangle + k\) map bounded sets onto bounded sets, neighborhoods onto neighborhoods, balls onto balls and open sets onto open sets.
Hyperplanes and Half spaces¶
Corresponding to a hyperplane \(\mathcal{H}\) in \(\RR^n\) (an \(n-1\)-flat), there exists a non-null vector \(a\) and a real number \(k\) such that \(\mathcal{H}\) is the graph of \(\langle a , x \rangle = k\). The vector \(a\) is orthogonal to \(PQ\) for all \(P, Q \in \mathcal{H}\). All non-null vectors \(a\) to have this property are normal to the hyperplane. The directions of \(a\) and \(-a\) are called opposite normal directions of \(\mathcal{H}\). Conversely, the graph of \(\langle a , x \rangle = k\), \(a \neq 0\), is a hyperplane for which \(a\) is a normal vector. If \(\langle a, x \rangle = k\) and \(\langle b, x \rangle = h\), \(a \neq 0\), \(b \neq 0\) are both representations of a hyperplane \(\mathcal{H}\), then there exists a real non-zero number \(\lambda\) such that \(b = \lambda a\) and \(h = \lambda k\). Obviously, we can find a unit norm normal vector for \(\mathcal{H}\). Each point \(P\) in space has a unique foot (nearest point) \(P_0\) in a Hyperplane \(\mathcal{H}\). Distance of the point \(P\) with vector \(p\) from a hyperplane \(\mathcal{H} : \langle a , x \rangle = k\) is given by
The coordinate \(p_0\) of the foot \(P_0\) is given by
Hyperplanes \(\mathcal{H}\) and \(\mathcal{K}\) are parallel if they don’t intersect. This occurs if and only if they have a common normal direction. They are different constant sets of the same linear functional. If \(\mathcal{H}_1 : \langle a , x \rangle = k_1\) and \(\mathcal{H}_2 : \langle a, x \rangle = k_2\) are parallel hyperplanes, then the distance between the two hyperplanes is given by
If \(\langle a, x \rangle = k\), \(a \neq 0\), is a hyperplane \(\mathcal{H}\), then the graphs of \(\langle a , x \rangle > k\) and \(\langle a , x \rangle < k\) are the opposite sides or opposite open half spaces of \(\mathcal{H}\). The graphs of \(\langle a , x \rangle \geq k\) and \(\langle a , x \rangle \leq k\) are the opposite closed half spaces of \(\mathcal{H}\). \(\mathcal{H}\) is the face of the four half-spaces. Corresponding to a hyperplane \(\mathcal{H}\), there exists a unique pair of sets \(\mathcal{S}_1\) and \(\mathcal{S}_2\) that are the opposite sides of \(\mathcal{H}\). Open half spaces are open sets and closed half spaces are closed sets. If \(A\) and \(B\) belong to the opposite sides of a hyperplane \(\mathcal{H}\), then there exists a unique point of \(\mathcal{H}\) that is between \(A\) and \(B\).
General Position¶
A general position for a set of points or other geometric objects is a notion of genericity. In means the general case situation as opposed to more special and coincidental cases. For example, generically, two lines in a plane intersect in a single point. The special cases are when the two lines are either parallel or coincident. Three points in a plane in general are not collinear. If they are, then it is a degenerate case. A set of \(n+1\) or more points in \(\RR^n\) is in said to be in general position if every subset of \(n\) points is linearly independent. In general, a set of \(k+1\) or more points in a \(k\)-flat is said to be in general linear position if no hyperplane contains more than \(k\) points.