(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".
In elementary geometry, a vector is simply the
difference between two points in space;
it is whatever has to be traveled to go from a given origin to some destination.
Etymologically, such a thing was perceived as "carrying"
the notion of distance between two points
(the radius from a fixed origin to a point).
The term vector 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.
As presented next, the term vector more
generally denotes an element of a linear space (vector space)
of an indefinite number of dimensions (possibly infinitely many)
over any scalar field (not necessarily the real numbers).
(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
|
(2010-04-23) Independent vectors.
Basis of a vector space.
The dimension is the largest possible number of independent vectors.
The modern definition of a vector space does not
involve the concept of dimension which had a towering
presence in the historical examples of vector spaces taken from
Euclidean geometry: A line has dimension 1, a plane has dimension 2,
"space" has dimension 3, etc.
The concept of dimension is best retrieved by introducing two complementary
notions pertaining to a set of vectors B.
- B is said to consist of
indenpendent vectors when all
nontrivial linear combinations
of the vectors of B are nonzero.
- B is said to generate E
when every vector of the space E is a linear combination of vectors of B.
A linear combinations
of vectors is a sum of finitely many of those
vectors, each multiplied by a scaling factor
(called a coefficient ).
A linear combination with
at least one nonzero coefficient
is said to be nontrivial.
If B generates E and
consists of independent vectors, then it's
called a basis of E.
Note that the trivial space {0} doesn't have a basis...
To prove that all nontrivial vector spaces have a basis
requires the Axiom of Choice
(in fact, the existence of a basis for any nontrivial
vector space is equivalent to the Axiom of choice ).
Dimension theorem for vector spaces :
A not-so-obvious statement is that all bases of E
can be put in one-to-one correspondence
with each other; they all have the same
cardinal (finite or not).
That cardinal is called the dimension of the space E:
dim (E)
Dimension
theorem for vector spaces
(2010-06-21)
Intersection and Sum of Subspaces.
A vector space included in another is called a subspace.
A subset F of a vector space E
is a subspace of E if and only if it is
stable by addition and scaling (i.e., the sum of two vectors of F
is in F and so is any vector of F
multiplied into a scalar).
It's an easy exercise to show that the intersection
FÇG of two subspaces
F and G is a subspace of E.
So is the sum F+G (defined as the set of all sums
x+y of a vector x from F and a vector
y from G).
Two subspaces of E for which
FÇG = {0} and
F+G = E are said to be supplementary.
Their sum is then called a direct sum and a
compact notation may be used which summarizes both previous relations:
E = F Å G
In the case of finitely many dimensions, the following relation holds:
dim ( F Å G )
= dim (F) + dim (G)
The generalization to nontrivial intersections is Grassmann's Formula :
dim ( F + G ) = dim (F) + dim (G)
-
dim ( F Ç G )
(2010-12-03)
Linear maps. Isomorphic vector spaces.
Two spaces are
isomorphic if there's a linear bijection between them.
A function f
which maps a vector space E into another space F
over the same field K is said
to be linear if it respects addition and scaling:
" x,y Î K,
" U,V Î E,
f ( x U + y V ) =
x f ( U ) +
y f ( V )
If such a linear function f is
bijective, its inverse is also
a linear map and the vector spaces
E and F are said to be isomorphic.
E » F
In particular, two vector spaces which have the same finite
dimension over the same field are necessarily isomorphic.
(2010-12-03)
Quotient E/H
of a vector space E by a subspace H
The equivalence classes (or residues) modulo H
can be called slices.
If H is a subspace of the vector space E, we
may consider the partition of E into sets
(let's call them slices) of the form
x+H. The
set of such slices is clearly a vector
space (scaling a slice or adding up two slices yields a slice).
This vector space is denoted E/H
and is called the quotient of E by H.
x+H denotes the set of all sums
x+h where h is an element of H.
E/H
is indeed the quotient of E
modulo the equivalence relation which defines as equivalent
two vectors whose difference is in H.
The canonical linear map which sends a vector
x of E to the slice x+H
is called the quotient map
of E onto E/H.
A vector space is always isomorphic to the
direct sum of one its subspaces
and its quotient by that same subspace:
E »
H Å E/H
Use this with H = ker ( f )
to prove the following fundamental theorem:
(2010-12-03)
Fundamental theorem of linear algebra.
Rank theorem.
A vector space is isomorphic to the
direct sum of the image and kernel
(French: noyau) of any linear function defined over it.
The image or range of a
linear function
f which maps a vector space E
to a vector space F is a subspace of F defined as follows:
im ( f ) = range ( f ) = f (E)
=
{ y Î F |
$
x Î E , f (x) = y }
The kernel (also called nullspace)
of f is the following subspace of E :
ker ( f ) =
null ( f ) =
{ x Î E |
f (x) = 0 }
The fundamental theorem of linear algebra states that
there is a subspace of E which is isomorphic to
f (E) and supplementary
to ker ( f ) in E.
This results holds for a finite or an infinite number of dimensions
and it's commonly expressed by the following isomorphism:
f (E) Å ker ( f )
» E
Proof :
This is a corollary of the above, since
f (E) and E / ker ( f )
are isomorphic because a bijective map between them is obtained
by associating uniquely f (x)
with the residue class x + ker ( f ).
Clearly, that association doesn't depend on the choice of x.
The above argument is itself an incarnation of the so-called
first
isomorphism theorem, as published by
Emmy Noether in 1927.
Restricted to vector spaces of finitely many dimensions,
the theorem amounts to the following
famous result (of great practical importance).
Rank theorem (or rank-nullity theorem) :
For any linear function f over a finite-dimensional space E, we have:
dim ( f (E) ) + dim ( ker ( f ) ) = dim ( E )
dim ( f (E) ) is called the rank of f.
The nullity of f is dim ( ker ( f ) ).
In the language of the matrices
normally associated with linear functions:
The rank and nullity
of a matrix add up to its number of columns.
The rank of a matrix A
is defined as the largest number of linearly independent columns
(or rows) in it.
Its nullity is the dimension of its nullspace (consisting, by definition, of
the column vectors x for which A x = 0).
Wikipedia :
Fundamental
theorem of linear algebra
|
Rank theorem
|
Rank of a matrix
(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").
(2007-11-06) 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).
(2009-09-03)
Two Flavors of Duality
Algebraic duality & topological duality.
In a vector space E,
a linear form is a linear function which maps every
vector of E to a scalar of the underlying
field K. The set of all linear forms is called the
algebraic dual of E.
The set of all continuous
linear forms is called the
[ topological ] dual of E.
With finitely many dimensions, the two concepts are identical
(i.e., every linear form is continuous)
not so with infinitely many dimensions.
An element of the dual (a continuous linear form)
is often called a covector.
Unless otherwise specified, we shall use the unqualified term dual
to denote the topological dual. We shall denote it E*
(some authors use E* to denote the algebraic dual and E'
for the topological dual).
The bidual E** of E is the dual of E*.
A canonical homomorphism exists which immerses E into E**
by viewing a vector
v of E as a linear form on E* which maps every
element f of E* to the
scalar f (v) :
v ( f ) = f (v)
If that canonical homomorphism is
a bijection, then E
is said to be reflexive and it is
routinely identified with its bidual E**.
E = E**
If E has infinitely many dimensions,
its algebraic bidual is never
isomorphic to it.
That's the main reason why the notion of topological
duality is retained. (Note that a
Hilbert space is always reflexive in the above sense,
even if it has infinitely many dimensions.)
Examples :
Consider the spaces
E = 
(
)
and
E* = .
The latter is the set of all real sequences and the former is the
set of real sequences with only finitely many nonzero values.
They happen to be [topological] duals of each other.
Dual space
(2009-09-03)
Tensorial Product and Tensors
E Ä F
is generated by tensor products.
Consider two vectors spaces
E and F over the
same field of scalars K.
For two covectors f and g
(respectively belonging to E* and F*) we may consider
a particular linear form denoted
f Ä g
and defined over the cartesian product
E´F via the relation:
f Ä g (u,v)
=
f (u) g (v)
The binary operator Ä
thus defined from (E*)´(F*)
to (E´F)*
is called tensor product.
(Even when E = F, the operator
Ä is not commutative.)
(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).
(2009-09-25)
On multi-dimensional objects that are "not vectors"...
To a mathematician, the juxtaposition (or
cartesian product )
of several vector spaces over the same field K is always
a vector space over that field (as component-wise definitions
of addition and scaling satisfy the above axioms).
When physicists state that some particular juxtaposition of
quantities (possibly a single numerical quantity by itself) is
"not a scalar", "not a vector" or "not a tensor"
they mean that the thing lacks an unambiguous and intrinsic definition.
Typically, a flawed vectorial definition would actually depend on
the choice of a frame of reference for the physical universe.
For example, the derivative of a scalar with respect to the first
spatial coordinate is "not a scalar" (clearly, that quantity depends on
what spatial frame of reference is chosen).
Less trivially, the gradient of a scalar is a physical covector
(of which the above happens to be one
covariant coordinate). Indeed,
the definition of a gradient specifies the same object (in dual
space) for any choice of a physical basis.
Some physicists routinely introduce (especially in the context
of General Relativity)
vectors as "things that transform like elementary displacements" and
covectors as "things that transform like gradients".
Their new students are thus expected to use a fairly complicated notion
(coordinate transformations) before the stage is set.
There can be an awkward moment there... Newbies will need several passes
through that intertwined logic before they "get it".
I find it far better to introduce the mathematical notion of a vector first.
Having easily absorbed that straight notion, the student may then be asked to consider
whether a particular rule defines a vector or not in complex physical situations.
(2007-08-21) The "Geometric Calculus" of Hestenes
Unifying some notations of mathematical physics...
Building on similitudes in several 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
