Descripción de las tablas

For every vector space V there exists an origin of the coordinate system with respect to which the vectors are defined. If we try to forget about the origin, we are left only with the set of points, which we show here as SV = {ψ_i}. Conversely, by choosing a point, say ψ₀ ∈ SV, we recover the vector space as V = SV − ψ₀ = {ψ_i− ψ₀ = ~ψ_i}.

Definition 1.1.10. A vector space without a specific choice of origin is called an affine space.

Then, the linear transformations of the vector space can be carried over to affine spaces as follows.

Definition 1.1.11. An affine transformation is a mapping on an affine space ˆA : SV → SV

which corresponds to a linear operator ˆA acting on the associated vector space V so that A(ψ) − ˆˆ A(φ) = ˆA( ~ψ − ~φ), (1.9)

for all ψ, φ ∈ SV.

Note that we have abused the notation to denote both a linear operation on V and an affine transformation on SV with the same symbol. We see that affine transformations are invariant under translations of the origin. Consequently, an affine transformation preserves points and lines, as well as the relative length of line segments of parallel lines, but not necessarily the angles. This is what one expects from linear operations on a vector space by forgetting about the origin. It turns out that any affine map can be represented on the associated vector space as ˆA → ˆA( ~ψ) + ~φ0 for some translation ~φ0. It is now natural to extend the primitive property of linear independence of vector spaces to affine spaces; see Fig. 1.1.

Definition 1.1.12. A subset of k points SV;k = {ψ_i}^k_i=1 in an affine space SV is called affinely independent if by defining an origin for the set, say ψ_J ∈ SV;k, the resulting k − 1 vectors {ψ_i− ψ_J} for i = 1 to j and i 6= J, are linearly independent.

By combining Definition 1.1.12with 1.1.8, every k affinely independent points will correspond to a k − 1 dimensional vector space.

Figure 1.1: Geometry of affine spaces. The representation of three linearly independent vectors underlying four affinely independent points in a three-dimensional vector (affine) space.

The pair (ψ₀, SV;3) with SV;3= {ψ₁, ψ₂, ψ₃} defines an affine frame. The combinations of points in SV;3 define a hyperplane which is a two-dimensional affine subspace.

Definition 1.1.13. The dimensionality of an affine space SV is defined to be the same as its underlying vector space V, i.e. dim SV = dim V.

In an n-dimensional affine space it is thus possible to choose at most n + 1 affinely independent points (see e.g. Fig.1.1). Inspired by this fact, we can also define the dimensionality of subsets of affine spaces as follows.

Definition 1.1.14. The dimensionality of a subset of points S in affine space is defined to be the maximum number of affinely independent points within S.

In contrast to the vector spaces possessing a basis set of vectors, in affine spaces such a notion does not exists per se, due to a lack of origin. It is, however, possible (and in fact more useful) to introduce the concept of a frame which also specifies the origin.

Definition 1.1.15. In an affine space SV over an n-dimensional vector space V the pair (ψ₀, SV;n), in which {ψ₀} ∪ SV;n = {ψ_i}ⁿ_i=0 is a set of n + 1 affinely independent points and ψ₀ plays the role of the origin, is called an affine frame.

Similar to the construction of affine spaces from linear vector spaces, one can construct an structure called the dual affine space from a dual vector space.

Definition 1.1.16. A dual vector space to V without a specific choice of origin is called a dual affine space to the corresponding affine space SV and is denoted by S_V^d.

Note that, the elements of a dual affine space are operations that map points from the affine space to the elements of the scalar field K. Moreover, the concepts of inner product and norm can also be carried over to the affine spaces.

Definition 1.1.17. An inner-product (normed) affine space is an affine space for which the underlying vector space is endowed with an inner-product (norm).

Note that, again we abuse the notation to denote the inner product of two points ψ, φ ∈ SV as hψ, φi = D ~ψ, ~φE

with respect to a specific origin. We also refer to the norm of a point ψ ∈ S_V as kψk =

with reference to a particular origin. It is thus clear that both inner product and norm of affine points depend on the choice of the origin. Naturally, we can also extend the notion of completeness over inner-product affine spaces.

Definition 1.1.18. Consider the normed affine space SV. Then, SV is a complete affine space if it contains the converging points of all Cauchy sequences.

With a little abuse of terminology, we also call a complete normed affine space a Banach space, and when the norm is induced by the inner product, a Hilbert space. Note that by using the term “complete affine space”, using the definition of normed affine spaces 1.1.17, we actually mean an affine space which is complete with respect to the norm defined on the associated vector space. Probably, the most familiar Banach space is the space of real numbers R with the absolute-value norm, which can be understood as the completion of the set of rational numbers.

It should be now evident that affine spaces can also be completed following Proposition 1.1.1.

Proposition 1.1.3. [2] Any normed affine space can be completed to a Banach space. There-fore, all inner-product affine spaces can be completed to Hilbert spaces.

Importantly, Riesz theorem1.1.1can also be applied to affine Hilbert spaces through which the dual affine space S_V^d of S_V can be fully characterized. Finally, as a consequence of Lemma1.1.3 and Proposition 1.1.2, we obtain the following results for affine spaces [3].

Lemma 1.1.4. All norms on a finite dimensional affine space S_V are equivalent, in the sense that, they are equivalent on the associated vector space V, as given in Lemma 1.1.3.

Proposition 1.1.4. All finite dimensional inner-product affine spaces are Banach spaces.

Similar to vector spaces, given a frame (ψ₀, S_V;n), every point ψ ∈ S_V can be written as

Consequently, we can describe the whole affine space as

SV = {ψ|∀x_i ∈ K, ψ =

form of coordinates for vectors. In fact, the mapping from points of SV to their Cartesian coordinates is a bijection providing an isomorphism from S_V to Kⁿ. However, it is sometimes easier to use the notation of Eq. (1.11) and define the following coordinates.

Definition 1.1.19. Given a point ψ in an n-dimensional affine space SV with the frame (ψ₀, SV;n), the vector of coefficients ˜x = (x₀, x₁, . . . , x_n) such that

is called the barycentric coordinates of the point ψ.

Combinations of affine points in the form (1.13) are called barycentric combinations which will be used frequently throughout this section.

Every component of the Cartesian coordinates x (and thus the barycentric coordinates ˜x) can be obtained from the corresponding vector in the underlying vector space. Equivalently, we can define a dual or reciprocal frame corresponding to the dual basis of affine space as below.

Definition 1.1.20. Given an affine frame (ψ₀, SH;n) with SH;n= {ψ_i}ⁿ_i=1 for an affine Hilbert space SH, the dual or reciprocal frame (ψ₀, S_H;n^d ) with S_H;n^d = {ψ^d_i}ⁿ_i=1 is a set of points in SH

such that ψ^d_i, ψ_j = δ_ij for all i and j.

Again, we emphasize that the inner product between points can only be understood in terms of the inner product on the underlying vector space with respect to a common origin, that is, ψ_i^d, ψ_j = ψ^d_i − ψ₀, ψ_j− ψ₀. The latter implies that ψ_i^d, ψ₀ = 0 for all i. Hence, given a point ψ its expansion coefficients in a given frame (ψ₀, SH;n) can be obtained using the dual frame (ψ₀, S_H;n^d ) as different frames for the n-dimensional affine Hilbert space SH. Every point of SH;n and ψ₀ of the former frame can be written in terms of the latter frame elements according to Eq. (1.14).

Thus, using the inner-product rule of the space SH, we have the following set of linear equations describing an arbitrary point ψ ∈ S_H in terms of the coordinates in (φ₀, S_H;n⁰ ),

in the frame (φ0, S_H;n⁰ ). Indeed, as expected, this is exactly the same as the description of the corresponding elements of the underlying (Euclidean) vector space using a Cartesian coordinate system.

Given a set of vectors it is always possible to construct a vector space containing those vectors by including the null vector and completing the set under addition of vectors and multiplication by scalars. The same approach can also be taken for affine sets of points resulting in a minimal container affine subspace, called the affine hull as per below.

Definition 1.1.21. The affine hull of an n-dimensional affine subset S ⊂ SV which contains a maximal set of affinely independent points S_n = {ψ_i}ⁿ_i=1 is the affine subspace generated by S_n⊂ S_V as

The minimality of this subspace is given in the following proposition.

Proposition 1.1.5. The affine hull S_aff of a subset S is the smallest affine subspace containing S.

Conversely, given a subspace of the vector space V, one can define the corresponding affine subspace of S_V. Such affine subspaces naturally possess their respective affinely independent points and frames.

Definition 1.1.22. Two affine subspaces are orthogonal if and only if their corresponding vector subspaces are orthogonal.

A particular class of such subspaces are affine planes; see Fig. 1.1.

Definition 1.1.23. In an n-dimensional affine space the linear combinations of k 6 n + 1 affinely independent points {ψ_i}^k−1_i=0 define a k − 1-dimensional plane given by

S_plane = {ψ|ψ ∈ SV, ∀x_i ∈ K, ψ =

Moreover, for k = n the n − 1-dimensional affine plane is called a hyperplane.

The plane S_plane is a subspace of the affine space SV and its generating points correspond to the frame (ψ₀, SV;k−1) with SV;k−1 = {ψ_i}^k−1_i=1 for this subspace. From Eq. (1.15), a point as a zero-dimensional object in an n-dimensional affine space is described by n linear equations in n coordinates. Similarly, a plane of dimension k in an n-dimensional affine space is described by n − k linear equations. The latter is more intuitive if we think of the k-dimensional plane as the set of all points restricted to be orthogonal to some n − k-dimensional subspace. As a result, when considering the set of equations (1.15) for such points, n − k of them correspond to the orthogonality constraint and the rest are redundant. For instance, in Eq. (1.17), assume that the n − k + 1 points {ψ_i}ⁿ_i=k that are not in the plane are affinely independent. Because they are

Figure 1.2: Geometrical representation of the two half-spaces. A hyperplane in a 3-dimensional affine space splits the space into two half-spaces. The frame (φ0, {φ}) defines the normal vector of the hyperplane. The yellow region is called the right-hand-side of the hyperplane. The points φ₁ and φ₂ are on the right- and left-hand-sides of the hyperplane and within the half-spaces S_+;c and S−;c, respectively.

not within S_plane we can assume that hψ_i, ψ_ji = 0 for all i ∈ {k, . . . , n} and j ∈ {1, . . . , k − 1}.

Then, the plane S_plane can be written as

S_plane= {ψ|ψ ∈ SV, ∀i ∈ {k, . . . , n}, hψ_i, ψi = 0}. (1.18) In the particular case of a hyperplane, only one linear equation is required and the hyperplane is uniquely determined by the set of points ψ satisfying the single equation

hφ, ψi = ψ^φ=

n−1

X

i=0

x_iψ^φ_i = 0,

n−1

X

i=0

x_i = 1, (1.19)

in n − 1 free parameters where φ is an affinely independent point orthogonal to the hyperplane.

Note also that ψ₀^φdetermines the offset of the hyperplane from the origin of the one-dimensional frame (φ0, {φ}). In the vector space picture, the vector φ − φ0 is called the normal vector of the hyperplane.

Corollary 1.1.1. Suppose that (ψ₀, SV;n) with SV;n = {ψ_i}ⁿ_i=1 is a frame for the n-dimensional affine space SV. Given a one-dimensional frame (φ₀, {φ}) for a one-dimensional affine subspace of SV, a hyperplane S_hp;c^φ in SV is given by

S_hp;c^φ = {ψ|ψ ∈ SV, x_i ∈ K, ψ =

n

X

i=0

x_iψ_i,

n

X

i=1

x_iψ_i^φ= c}, (1.20)

in which ψ_i^φ = hφ, ψ_ii is the component of the point ψ_i in the frame (φ₀, {φ}), and c ∈ K

determines the offset from the origin. The vector φ − φ0 is called the normal vector of the hyperplane (see Fig. 1.2).

It is now very intuitive to define the following geometrical objects which will be useful later on.

Definition 1.1.24. Given a hyperplane S_hp;c^φ with normal vector φ − φ₀, it splits the whole affine space SV into two subsets given by

S_+;c = {ψ|ψ ∈ S_V, x_i ∈ K, ψ =

S_+;c and S−;c are called (closed) half-spaces. The positive and negative half-spaces are said to be on the right-hand-side and left-hand-side of the hyperplane, respectively (see Fig. 1.2).

Evidently, the intersection of the two half spaces is the hyperplane, S_+;c∩ S−;c = S_hp;c, while their union gives the whole affine space, S_+;c∪ S−;c= SV.