282
APPENDIX A Mathematical Background
"gradient", for instance, and will differ by the extent of their respective
domains.
A.2 IMPORTANT STRUCTURES
A.2.1 Groups
Groups are important, not only because many mathematical structures
like linear space, algebra, etc., are first and foremost groups, with added
features, but as a key to
symmetry.
A group
is a set equipped with an associative binary operation, with
a neutral element and for each element, an inverse. Examples: the group
Z of relative integers, the regular matrices of some definite order, etc.
As these two examples show, the group operation may or may not
be commutative, hence a notational schism. Commutative, or
Abelian,
groups, like Z, are often denoted additively. But in the general case, the
operation is called a product, denoted without any symbol, by~ simple
juxtaposition, the neutral element is 1, and the inverse of g is g .
A group G
acts
on a set X if for each g ~ G there is a map from X
to X, that we shall denote by ~(g), such that ~(1) is the identity map,
and ~(gh)= ~(g) 0 ~(h). 19 Observe, by taking h = g-l, that ~(g) must be
bijective, so ~(g) is a
permutation
of X. The set {~(g) : g ~ G} is thus a
group of permutations, 2° the group law being composition of maps. Let's
denote this set by ~(G).
The same abstract group can act in different ways on various related
geometric objects: points, vectors, plane figures, functions, fields, tensors,
etc. What counts with groups is their actions. Hence the importance of
the related vocabulary, which we briefly sketch.
The action
is faithful,
or
effective,
if ~(g) = 1 implies g = 1. (Informally,
an action on X is effective if all group elements "do something" on X.)
In that case, G and ~(G) are isomorphic, and ~(G) can be seen as a
"concrete" realization of the "abstract" group G. This justifies writing
gx, instead of ~(g)x, for the image of x by ~(g). The
orbit
of x under
the action of G is the set {gx : g ~ G} of transforms of x. Points x and
y are in the same orbit if there exists some group element g that
19This is called an action
on the left,
or
left action,
as opposed to a
right action,
which
would satisfy ~(gh) = ~(h) o ~(g), the other possible convention. A non-Abelian group can
act differently from the left and from the right, on the same set. All our group actions will
be on the left.
2°A subgroup of the "symmetric group" S(X), which consists of
all
permutations on X,
with composition as the group law.
A.2 IMPORTANT STRUCTURES 283
transforms x into y. This is an equivalence relation, the classes of
which are the orbits. If all points are thus equivalent, i.e., if there is a
single orbit, one says the action is
transitive. The isotropy
group (or
stabilizer,
or
little group)
of x is the subgroup G x = {g ~ G : gx = x} of elements of
G that fix x. A transitive action is
regular
if there are no fixed points,
that is, G x = 1 for all x (where 1 denotes the trivial group, reduced to
one element).
In the case of a regular action, X and G look very much alike, since
they are in one-to-one correspondence. Can we go as far as saying they
are identical? No, because the group has more structure than the set it
acts upon. For a simple example, imagine a circle. No point is privileged
on this circle, there is no mark to say "this is the starting point". On the
other hand, the group of planar rotations about a point (where there
is a
distinguished element, the identity transform) acts regularly on this circle.
Indeed, the circle and this group (traditionally denoted SO 2)
can
be
identified. But in order to
do
this identification, we must select a point of
the circle and decide that it will be paired with the identity transform.
The identification is not canonical, and there is no group structure on the
circle before we have made such an identification.
The concept of
homogeneous space
subsumes these observations. It's
simply a set on which some group acts transitively and faithfully. If,
moreover, the little group is trivial (regular action), the only difference
between the homogeneous space X and the group G lies in the existence
of a distinguished element in G, the identity. Selecting a point O in X
(the origin)and then identifying gO with gmhence O in X with 1 in
Gmprovides X with a group structure.
So when homogeneity is mentioned, ask what is supposed to be
homogeneous (i.e., ask what the elements of X are) and ask about the
group action. (As for isotropy and other words in tropy, it's just a
special kind of homogeneity, where the group has to do with rotations in
some way.)
A.2.2 Linear spaces: V n, A~
I don't want to be rude by recalling what a
vector space
(or
linear space)
is,
just to stress that a vector space V is already a group (an Abelian one),
with the notion of scalar 21 multiplication added, and appropriate axioms.
The
span
v{v i : i~ y} of a family of vectors of V is the set of all
weighted sums ~i~ j i vi, with scalar coefficients ff only a finite
21Unless otherwise specified, the field of scalars is IR.
284 APPENDIX A Mathematical Background
number of which are nonzero (otherwise there is nothing to give sense to
the sum). This span, which is a vector space in its own right, is a
subspace
of V. A family is linearly
independent
if the equality
~iO~ 1 V i --
0 forces
all ~ = 0. The highest number of vectors in a linearly independent
family is the
dimension
of its span, if finite; otherwise we have an
infinite
dimensional
subspace. The notion applies as a matter of course to the
family of all vectors of V. If the dimension dim(V) of V is n, one may,
by picking a basis (n independent vectors el, ..., e), write the generic
vector v as v 1 e 1 +... + v nen, hence a one-to-one correspondence v
{v 1, ..., v n} between v and the n-tuple of its
components.
So there is an
isomorphism (non-canonical) between V and IR n, which authorizes one
to speak of
the
n-dimensional real vector space. That will be denoted
V n. Don't confuse V n and IR n, however, as already explained. In an
attempt to maintain awareness of the difference between them, I use
boldface for the components, 22 and call the n-tuple v = {v 1, ..., v n} they
form, not only a vector (which it is, as an element of the vector space
IRn), but a vector. Notation pertaining to IR n will as a rule be in boldface.
A relation r = {V, W, R}, where V and W are vector spaces is
linear
if the graph R is a vector space in its own right, that is, a subspace of the
product V x W. If the graph is functional, we have a
linear map.
Linear
maps s : V ~ W are thus characterized by s(x + y) = s(x) + s(y) and
s(Lx) = Ls(x) for all factors. Note that dora(s) and cod(s) are subspaces
of V and W.
Next, affine spaces. Intuitively, take V, forget about the origin, and
what you have got is
An,
the n-dimensional affine space. But we are
now equipped to say that more rigorously. A vector space V, considered
as an additive group, acts on itself (now considered just as a set) by the
mappings rt(v) = x --~ x + v, called
translations.
This action is transitive,
because for any pair of points {x, y}, there is a vector v such that y = x
+ v, and regular, because x + v ~ x if v ~ 0, whatever x. The structure
formed by V as a set 23 equipped with this group action is called the
affine space A associated with
V. Each vector of V has thus become a
point of A, but there is nothing special any longer with the vector 0, as a
point in A.
More generally, an
affine space
A is a homogeneous space with
respect to the action of some vector space V, considered as an additive
22At
least, when such components can be interpreted as degrees of freedom, in the
context of the finite element method. Our DoF-vectors are thus vectors. (Don't expect
absolute consistency in the use of such conventions, however, as this can't be achieved.)
23Be well aware that V is first
stripped
of its operations, thus becoming a mere set, then
refurbished
with this group action, to eventually become something else, namely A.
A.2 IMPORTANT STRUCTURES 285
group. By selecting a point 0 in A
to play origin, we can identify vector
v of V with point 0+v of A. But
there may be no obvious choice for an
origin. For example [Bu], having
selected a point x and a line g through
x in 3D space, all planes passing
through x and not containing ~ form
an affine space (inset). None of them
0
is privileged, and the group action is not
obvious. 24
For an easier example,
consider a subspace W of some vector space V, and define an equivalence
r by u r v ,=~ v - u ~ W. Equivalence classes have an obvious affine
structure (W acts on them regularly by v ~ v + w) and are called
affine
subspaces
of V,
parallel
to W. Of course, no point of an affine subspace
qualifies more than any other as origin.
Remark A.3. The latter is not just any equivalence relation, but one
which is compatible as with the linear structure: if x r y, then Xx r Ky,
and (x+z) r(y+z). This way, the
quotient X/r is a vector space. Now
if one wants to select a representative
section, it makes sense to preserve this
compatibility, by requesting this
section to be a vector subspace U of
V (inset, to be compared with Fig. A.9),
which is said to be
complementary
with
respect to W. Then each v ~ V can
V J /i ~r
~
~
V/r
uniquely be written as v = u + w, with u ~ U and w ~ W. Again, don't
confuse the quotient V/r with the complement U, although they are
isomorphic. 0
Affine space is perhaps the most fundamental example of homoge-
neous space. From a philosophical standpoint, the fact that we chose to
do almost all our applied physics in the framework provided by
A 3
(plus, when needed, a time parameter) reflects the
observed
homogeneity
of the space around us.
24Take a vector u parallel to g, and two parallels 0' and g" to u, distinct from g.
They pierce plane x at x' and x". The "translation" associated with {)~', K"} ~ IR 2 is the
mapping x ~ {the plane determined by 0, x' + )~'u, x" + ~" u}.
25Note the importance of this concept of compatibility between the various structures
put on a same set.
286 APPENDIX A Mathematical Background
What you can do on
VECTORS
may not be doable on
POINTS.
Indeed,
the product Kx is meaningless in an affine space: What makes sense is
barycenters.
The barycenter of points x and y with respective weights
~, and 1 - ~ is x + K(y- x). Generalizing to n points is easy. Affine
independence, dimension of the affine space, and affine subspaces follow
from the similar concepts as defined about the vector space.
Barycentric
coordinates
could be discussed at this juncture, if this had not already
been done in Chapter
3. 26
Affine relations
are characterized by affine graphs. If the graph is
functional, we have an
affine map.
Affine maps
on
A n are those that are
linear with respect to the (n + 1)-vector of barycentric coordinates.
Affine subspaces
are the pre-images of affine maps. Affine subspaces of a
vector space are of course defined as the affine subspaces of its associate.
The sets of solutions of equations of the form Lx = k, where L is a linear
map (from V n to V m, m < n) and k a vector, are affine subspaces, and
those corresponding to the same L and different k's are parallel. The
one corresponding to k = 0 (called
kernel
of L, denoted ker(L)) is the
vector subspace parallel to them all.
If x is a point in affine space A, vectors of the form y- x are called
vectors at
x. They form of course a vector space isomorphic with the
associate V, called
tangent space at
x, denoted T x. (In physics, elements
of V are often called
free vectors,
as opposed to
bound vectors,
which are
vectors "at" some point.) The tangent space to a curve or a surface that
contains x is the subspace of T x formed by vectors at x which are
tangent to this curve or surface. Note that vector fields are maps of type
POINT -o BOUND_VECTOR,
actually, with the restriction that the value
of v at x, denoted v(x), is a vector at x. The distinction between this
and a
POINT --~ FREE_VECTOR
map, which may seem pedantic when
the point spans ordinary space, must obviously be maintained in the case
of fields of tangent vectors to a surface.
A convex set
in an affine space is a part C such that
(x~C and y~C)<=> Kx+(1-~,)y~C V~,~ [0,1].
Affine subspaces are convex. The intersection of a family of convex sets
is a convex set. The
convex hull
of a part K is the intersection of all
convex sets containing K, and thus the smallest of such sets. It coincides
with the union of all barycenters, with nonnegative weights, of pairs of
points of K.
26One
may--but it's a bit more awkward than the previous approachmdefine affine
spaces ab initio, without first talking of vector spaces, by axiomatizing the properties of the
barycentric map,
which sends {x, y, K} to Kx + (1 - K)y.
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