Vectors and Vector Spaces| Vectors and Vector Spaces |
In General > s.a. vector calculus; vector field.
* Versions: An arrow between points
on \(\mathbb R\)n; An infinitesimal
displacement in a manifold; An element of a vector space; A contravariant rank-1 tensor,
i.e., one of type (1, 0).
* In physics: Example of the types
of vectors used in physics are Gibbs' three-vectors, Minkowski four-vectors, complex
spinors in quantum mechanics, quaternions used to describe rigid body rotations and
vectors defined in Clifford geometric algebra.
@ References: Weinreich 98 [geometrical];
Lesche et al AJP(92)jun [dual/covector];
Fleisch 11 [II];
Chappel et al IEEE(16)-a1509 [different vector formalisms, historical perspective].
@ Covariant and contravariant: Schmidt AJP(97)nov-gq;
Kumar a1002 [pedagogical].
Vector Space
> s.a. Bilinear Form; Flag;
Hahn-Banach Theorem; Subspace.
$ Def: A set X with two
binary operations +: X × X → X and
· : K × X → X, where K is
a field (usually \(\mathbb R\) or \(\mathbb C\)), satisfying some properties.
* Linearly independent vectors: A set
of two or more vectors in a vector space is linearly independent if the only linear
combination of the vectors in the set that equals 0 is the one with all coefficients
equal to zero; Alternatively, none of the vectors is a linear combination of the others.
* New vector spaces out of old:
> see Direct Sum (of R-modules),
direct or tensor product.
* Ordered vector space: A vector space
with a partial order < satisfying simple compatibility conditions with the addition
and scalar multiplication.
* Relationships: A vector space structure is
stronger than an affine structure; > s.a. affine structure.
@ References: Halmos 74.
> Online resources:
see MathWorld page;
Wikipedia page.
Topological Vector Space > s.a. Banach Space;
Fréchet Space; norm.
$ Def: A vector space X over
a topological field K such that the two operations are continuous.
* Examples: Any metric
space or normed vector space, including Banach spaces and Hilbert spaces.
* Remark: On a real or complex
vector space there is a unique, natural topology that makes it a topological vector
space (by the Tychonoff theorem).
* Locally convex: A tvs which admits
a topological base of convex sets; A real vector space with a (Hausdorff) topology
generated by a family of seminorms.
@ References: Bourbaki 66;
Kelley 76.
> Online resources:
see MathWorld page;
Wikipedia page.
Vector Algebra in Euclidean Space
* Scalar (dot) product: X · Y
= XY cosθ = ∑i
XiYi.
* Vector (cross) product:
(Xx, Xy,
Xz) ×
(Yx, Yy,
Yz)
= (XyYz−XzYy,
XzYx−XxYz,
XxYy−XyYx),
which can also be expressed in determinant form, or as (X ×
Y)i
= εijk
XjYk.
@ References: Silagadze JPA(02) [7D generalization of vector product].
Inner Product > s.a. distance;
formulations of quantum mechanics; hilbert space;
Schwarz Inequality; types of symplectic structures.
$ Def: A map \(\langle\ ,\ \rangle:
V\times V\to {\mathbb R}\ ({\mathbb C})\), where V is a vector space
over \(\mathbb R\) (\(\mathbb C\)), which is bilinear (sesquilinear), Hermitian,
and positive definite.
* Examples: For probability
distributions, the information metric [@ Groisser & Murray
dg/96];
For complex functions,
(A, B):= ∫ dz A*(z) B(z) F(z) , F(z) > 0 for all z .
* And other structure:
It can always be used to define a norm by ||x||:= \(\langle\)x,
x\(\rangle\)1/2, and thus a distance.
* Inner product space: A pair (V,
\(\langle\) , \(\rangle\)) as above; An example is any Hilbert space.
@ General references:
Dvurečenskij LMP(01) [criterion for completeness];
Horváth JGP(10)
[semi-definite and indefinite inner products, generalized Minkowski spaces].
@ Partial inner product spaces:
Antoine et al AMP-a1203 [categorical aspects];
Antoine & Trapani JPA(13)-a1210.
Related Concepts > see Bivector;
Riesz Space [vector lattice].
* Axial vector or pseudovector:
A vector whose definition implies that the vector corresponding to a mirror reflected
situation is the opposite of the mirror image of the original vector; > s.a.
stochastic quantization [axial vector gauge theory];
Wikipedia page.
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send feedback and suggestions to bombelli at olemiss.edu – modified 10 nov 2018