Vector Space

A Vector Space is a metric space composed of a vector set and a vector operation set (that abide by the closure, addition and multiplication axioms).

• AKA: [math]\displaystyle{ V }[/math], n-Dimensional Space.
• Context:
  • It must abide by the Vector Closure Axioms.
    • Addition Closure; [math]\displaystyle{ ∀\mathbf{v},\mathbf{w} ∈ V }[/math], [math]\displaystyle{ ∃ \mathbf{x} ∈ V }[/math] called the sum of [math]\displaystyle{ \mathbf{v} }[/math] and [math]\displaystyle{ \mathbf{w} }[/math], that can also be denoted by [math]\displaystyle{ \mathbf{v}+\mathbf{w} }[/math].
    • Real Number Multiplication Closure; [math]\displaystyle{ ∀\mathbf{v} ∈ V }[/math] and [math]\displaystyle{ ∀α ∈ \mathbb{R} }[/math], [math]\displaystyle{ ∃ }[/math] an element in [math]\displaystyle{ V }[/math] called the product of [math]\displaystyle{ α }[/math] and [math]\displaystyle{ \mathbf{v} }[/math], denoted by [math]\displaystyle{ α\mathbf{v} }[/math].
  • It must abide by the Vector Addition Axioms.
    • Vector Addition Commutativity Axiom; [math]\displaystyle{ ∀\mathbf{v},\mathbf{w} ∈ V, \mathbf{v}+\mathbf{w} = \mathbf{w}+\mathbf{v} }[/math].
    • Vector Addition Associativity Axiom; [math]\displaystyle{ ∀u,\mathbf{v},\mathbf{w} ∈ V, \mathbf{u}+(\mathbf{v}+\mathbf{w}) = (\mathbf{u}+\mathbf{v}) + \mathbf{w} }[/math].
    • Zero Vector Axiom; [math]\displaystyle{ ∃ }[/math] an element [math]\displaystyle{ 0 ∈ V }[/math], such that [math]\displaystyle{ \mathbf{u} + 0 = \mathbf{u}, ∀ \mathbf{u} ∈ V }[/math].
    • Inverse Vector Axiom; [math]\displaystyle{ ∀\mathbf{v}∈V }[/math], there exists element [math]\displaystyle{ -\mathbf{v} }[/math] such that [math]\displaystyle{ \mathbf{v}+ -\mathbf{v}=0 }[/math].
  • It must abide by the Vector Scalar Multiplication Axiom.
    • Vector Scalar Multiplication Associativity Axiom; [math]\displaystyle{ ∀ \mathbf{u} ∈ V }[/math] and [math]\displaystyle{ ∀α,β ∈ \mathbb{R}, α(β\mathbf{v}) = (αβ)\mathbf{v} }[/math].
    • Scalar Multiplication Distributive Axiom for Vector Addition; [math]\displaystyle{ ∀\mathbf{v},w ∈ V }[/math] and [math]\displaystyle{ ∀α∈\R, α(\mathbf{v}+\mathbf{w}) = α\mathbf{v}+ α\mathbf{w} }[/math].
    • Scalar Multiplication Distributive Axiom for Scalar Addition; [math]\displaystyle{ ∀\mathbf{v}∈ V }[/math] and [math]\displaystyle{ ∀ α,β ∈ \mathbb{R},atj\gt , (α + β)\mathbf{v} = α\mathbf{v} + β\mathbf{v} }[/math].
  • It can abide by an Identity Axiom; [math]\displaystyle{ ∀ \mathbf{u} ∈ V, 1\mathbf{u}=\mathbf{u} }[/math].
  • It can range from being a Normed Vector Space (with a vector length function) to being a Non-Normed Vector Space.
  • It can range from being a Binary Vector Space, to being an Integer Vector Space, to being a Real Vector Space, to being a Complex Vector Space.
  • It can range from being a Finitely-Dimensional Vector Space (2D, |3D, 4D, ... n-Dimensional Finite Space) to being an Infinitely-Dimensional Vector Space.
• Example(s):
  • The set [math]\displaystyle{ V=\{ (x, y, z)^T \in \mathbb{R}^3|x+y=0\} }[/math] is a vector space.

    It can also be written as [math]\displaystyle{ V=\{ (x, -x, z)^T \in \mathbb{R}^3 \} }[/math]. The basis for [math]\displaystyle{ V }[/math] is [math]\displaystyle{ \left\lbrace \begin{bmatrix}1 \\-1 \\0 \end{bmatrix}, \begin{bmatrix}0 \\0 \\1 \end{bmatrix} \right\rbrace }[/math]. The dimension of V is 2 (number of elements in basis of [math]\displaystyle{ V }[/math]).

  • The set [math]\displaystyle{ V=\{ (x, y, z)^T \in \mathbb{R}^3|4x+z=0, 3y=z\} }[/math] is a vector space.

    It can also be written as [math]\displaystyle{ V=\{ (x, -\frac{4}{3} x, -4x)^T \in \mathbb{R}^3 \} }[/math]. The basis for [math]\displaystyle{ V }[/math] is [math]\displaystyle{ \left\lbrace \begin{bmatrix}1 \\-\frac{4}{3} \\-4 \end{bmatrix} \right\rbrace }[/math]. The dimension of V is 1 (number of elements in basis of [math]\displaystyle{ V }[/math]).

  • a Word Vector Space (for word vectors).
  • a DNA Vector Space (for DNA vectors).
  • an Inner Product Space.
  • a Semantic Vector Space.
  • …
• Counter-Example(s):
  • The set [math]\displaystyle{ V=\{ (x, y, z)^T \in \mathbb{R}^3|x\geqslant 0, y=-4z\} }[/math] is not a vector space.
  • The set [math]\displaystyle{ V=\{ (x_1, x_2,\dots,x_n)^T \in \mathbb{R}^n\mid |x_i|\leqslant 1; j=1,2,\dots,n\} }[/math] is not a vector space.
  • a Tuple Space.
• See: n-Dimensional Euclidean Space, Metric Space, Affine Space, Banach Space, Topological Space, Linear Algebra, Functional Analysis, Abstract Space.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/vector_space Retrieved:2015-1-9.
  • A vector space is a mathematical structure formed by a collection of elements called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.

    An example of a vector space is that of Euclidean vectors, which may be used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are best thought of as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows.

    Vector spaces are the subject of linear algebra and are well understood from this point of view since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. A vector space may be endowed with additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide whether a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional structure, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory.

    Historically, the first ideas leading to vector spaces can be traced back as far as 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs.

    Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations; offer a framework for Fourier expansion, which is employed in image compression routines; or provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra.

2009a

• (Wiktionary, 2009) ⇒ http://en.wiktionary.org/wiki/vector_space
  • 1. (mathematics) A type of set of vectors that satisfies a specific group of constraints. A vector space is a set of vectors which can be linearly combined.
  • 1. (vector space over the field F - linear algebra) A set V, whose elements are called "vectors", together with a binary operation + forming a module (V,+), and a set F* of bilinear unary functions f*:V→V, each of which corresponds to a "scalar" element f of a field F, such that the composition of elements of F* corresponds isomorphically to multiplication of elements of F, and such that for any vector v, 1*(v) = v.
    • Any field F is a one-dimensional vector space over itself.
    • If V is a vector space over F and [math]\displaystyle{ S }[/math] is any set, then V^S={f|f:S -> V} is a vector space over F, and Dim( V^S ) = Card(S) Dim(V).
    • If V is a vector space over F then any closed subset of V is also a vector space over F.
    • The above three rules suffice to construct all vector spaces.

2009b

• http://www.cs.caltech.edu/~westside/quantum-glossary.htm
  • a nonempty set of objects, called elements, that satisfy the following ten axioms: Let V denote a vector space, Closure axioms Axiom 1. ...

2009c

• http://www.quercus-sys.com/home/flt/flt10.htm
  • A mathematical system consisting of a set of points ("vectors") that form an abelian group and which allow for "multiplication" ...

2009d

• http://www.definecynical.net/viewtopic.php
  • A vector space over a field F is a set V that satisfied the following properties: i) There exists vector addition VxV->V such that: i) a+b=b+a ...

1997

• (Luenberger, 1997) ⇒ David G. Luenberger. (1997). “Optimization by Vector Space Methods." Wiley Professional. ISBN:047118117X

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