(2006-05-07) Etymology
Vectors were so named because they "carry" the distance from the origin.
In medical and other contexts,
"vector" is synonymous with "carrier".
The etymology is that of "vehicle": The latin verb
vehere means "to transport".
Before the concept was generalized beyond recognition,
a mathematical vector was simply understood as the "difference"
between two points in space.
In that basic meaning, a vector is whatever has to be traveled to go from
a given origin to a destination.
What was etymologically important was that such things were perceived as "carrying"
conceptually the notion of distance between two points,
the "radius" from a fixed origin to an arbitrary
point.
The term vector thus started out its mathematical life as part of
the French locution "rayon vecteur" (radius vector).
The whole expression is still used to identify a point in
ordinary (Euclidean) space, as seen from a fixed origin.
The simpler term "vector" has been generalized to spaces of an indefinite
number of dimensions (possibly infinitely many) over any scalar field
(not necessarily the real numbers) with the abstract definition presented
next.
(2006-03-28) Vector Space over a Field K
Vectors can be added, subtracted or scaled.
The scalars form a field.
A scalar is an element of the
field K.
A vector space E is a set with a well-defined
addition (the sum U+V of two vectors is a vector)
and multiplication by a scalar (a scaled vector x U
is still a vector) obeying the following rules:
-
(E, + ) is an Abelian group.
This is to say that the addition of vectors is an associative and commutative
operation and that subtraction is defined as well
(i.e., there's a zero vector,
neutral for addition, and every vector has an
opposite which yields zero when added to it).
-
Scaling is compatible with arithmetic on the field
K :
|
"xÎK,
"yÎK,
"UÎE,
"VÎE,
| |
(x + y) U =
x (U + V) =
(x y) U =
1 U =
|
x U + y U
x U + x V
x (y U)
U
|
(2007-11-06) Normed Vector Spaces & Banach Spaces
Banach Spaces are complete normed vector spaces.
-
You can't do a thing
with a space that's not complete.
Laurent Schwartz (1915-2002) lecturing in 1977.
Vector spaces are usually endowed with a function (called norm)
which associates to any vector V
a real number ||V||
(called the norm or the
length of V) such that the following
properties hold:
- ||V|| is positive for any nonzero vector V.
- ||lV|| =
|l| ||V||
- || U + V || ≤ || U + V ||
In this, l
is a scalar and
|l| denotes what's called a
valuation on the field of scalars
(a valuation is a special type of one-dimensional norm;
the valuation of a product is the product of the valuations of its factors).
Examples of valuations include the absolute value
of real or complex numbers and the
p-adic metric of p-adic numbers.
A Banach space is a normed vector space which is complete
(which is to say that every Cauchy sequence
in it converges).
Such structures are named after the Polish mathematician
Stefan Banach (1892-1945).
Arguably, they are the main backdrop for what's called analysis,
the branch of mathematics which revolves around the very notion of
limit
(it would be hazardous to discuss limits in a space that's not complete).
(2006-03-28) Module over a Ring K
A vectorial structure where division by a scalar isn't "well defined".
A module obeys the same basic rules as a
vector space,
but its scalars are only required to
form a ring;
a nonzero scalar need not have a reciprocal...
A module over K may be called a K-module.
For example,
is a
-module.
This is to say that the rationals form a module over the integers
(this particular example gave birth to the concept of an
"injective module").
(2007-04-30) Algebra over a Field K
An internal product among vectors turns a vector space into an algebra.
A so-called algebra is the structure obtained when
an internal multiplication is defined on the vector space E
(the product of two
vectors being a vector) which is both scalable and
distributive (over addition). That is to say:
|
"xÎK,
"yÎK,
"UÎE,
"VÎE,
"WÎE,
| |
(x y) (U V) =
U (V + W) =
(V + W) U =
|
(x U) (y V)
U V + U W
V U + W U
|
An algebra is also sometimes understood to be associative:
U (V W) = (U V) W
However, it's better to speak of an
associative algebra whenever applicable.
The octonions are an example of a
non-associative algebra.
Octonions form only an alternative algebra,
which is to say that:
U (V V) = (U V) V
and
U (U V) = (U U) V
The weakest form of associativity is power-associativity
which states that:
U (U U) = (U U) U
(2007-04-30) Clifford algebras over a Field K
Unital associative algebras endowed with a quadratic form.
Those structures are named after the British geometer and philosopher
William
Clifford (1845-1879).
(2007-08-21) The "Geometric Algebra" of David Hestenes
A proposal for unifying some notations of mathematical physics.
Building on obvious similitudes in several related areas of mathematical
physics, David Hestenes has been advocating a denotational unification
which has gathered a few enthusiastic followers.
The approach is called Geometric Algebra by its proponents.
It's unrelated to the abstract field of Algebraic Geometry
(which has been at the forefront of mainstream mathematical research for decades).
Geometric Calculus
by
David Hestenes
(Oersted
Medal Lecture, 2002).
Geometric Algebra Research
Group (at the Cavendish Laboratory ).
Excursions en algèbre
géométrique by Georges Ringeisen.
Wikipedia:
Geometric Algebra
