Vector Space

A Vector Space is a mathematical structure consisting of a vector set together with vector operations of vector addition and scalar multiplication that satisfy specific vector space axioms.

• AKA: Linear Space, [math]\displaystyle{ V }[/math], n-Dimensional Space.
• Context:
  • It can satisfy Vector Closure Axioms for both vector addition operations and vector scalar multiplication operations.
  • It can satisfy Vector Addition Axioms including vector addition commutativity axiom, vector addition associativity axiom, zero vector axiom, and inverse vector axiom.
  • It can satisfy Vector Scalar Multiplication Axioms including scalar multiplication associativity axiom and distributivity axioms.
  • It can support Linear Transformation Operations that preserve its vector space structure.
  • It can define Linear Independence Relations among its vector elements.
  • It can establish Vector Space Spans through linear combinations of vector subsets.
  • It can determine Vector Dimensions through its basis vector set.
  • ...
  • It can often serve as Mathematical Frameworks for linear algebra subject area.
  • It can often provide Foundations for functional analysis subject area.
  • It can often model Physical Systems through state space representations.
  • It can often support Optimization Algorithms including constrained optimization algorithms.
  • It can often enable Machine Learning Models through vectorized learning records.
  • ...
  • It can range from being a Finite-Dimensional Vector Space to being an Infinite-Dimensional Vector Space, depending on its vector space dimensional characteristic.
  • It can range from being a Real Vector Space to being a Complex Vector Space, depending on its vector space scalar field.
  • It can range from being a Normed Vector Space to being a Non-Normed Vector Space, depending on its vector space metric structure.
  • It can range from being a Complete Vector Space to being an Incomplete Vector Space, depending on its vector space Cauchy sequence behavior.
  • ...
  • It can contain Hyperplanes as affine subsets defining decision boundaries.
  • It can support Vector Addition Operations and vector subtraction operations.
  • It can normalize Vectors to produce unit vectors and normalized vectors.
  • It can be embedded in Larger Vector Spaces through vector space embeddings.
  • ...
• Examples:
  • Euclidean Vector Spaces, such as:
    • Real Coordinate Spaces, such as:
      • 2D Space ([math]\displaystyle{ \mathbb{R}^2 }[/math]) for vector space planar geometry.
      • 3D Space ([math]\displaystyle{ \mathbb{R}^3 }[/math]) for vector space spatial geometry.
      • 4D Space ([math]\displaystyle{ \mathbb{R}^4 }[/math]) for vector space spacetime modeling.
      • n-Dimensional Finite Space ([math]\displaystyle{ \mathbb{R}^n }[/math]) for vector space finite-dimensional analysis.
    • Complex Coordinate Spaces, such as:
      • Complex Plane ([math]\displaystyle{ \mathbb{C} }[/math]) for vector space complex analysis.
      • n-Dimensional Complex Space ([math]\displaystyle{ \mathbb{C}^n }[/math]) for vector space quantum states.
  • Function Vector Spaces, such as:
    • Continuous Function Spaces, such as:
      • [[C([a,b]) Space]] for vector space continuous function analysis.
      • C^∞ Space for vector space smooth function analysis.
    • Integrable Function Spaces, such as:
      • L^p Spaces for vector space measure theory.
      • Sobolev Spaces for vector space differential equations.
  • Specialized Vector Spaces, such as:
    • Inner Product Spaces for vector space geometric structure.
    • Normed Vector Spaces for vector space distance measurement.
    • Word Vector Spaces for vector space natural language processing.
    • DNA Vector Spaces for vector space genomic analysis.
    • Semantic Vector Spaces for vector space meaning representation.
    • State Spaces for vector space system modeling.
  • Constrained Vector Spaces, such as:
    • The set [math]\displaystyle{ V=\{ (x, y, z)^T \in \mathbb{R}^3|x+y=0\} }[/math] with vector space basis [math]\displaystyle{ \left\lbrace \begin{bmatrix}1 \\-1 \\0 \end{bmatrix}, \begin{bmatrix}0 \\0 \\1 \end{bmatrix} \right\rbrace }[/math] and vector space dimension 2.
    • The set [math]\displaystyle{ V=\{ (x, y, z)^T \in \mathbb{R}^3|4x+z=0, 3y=z\} }[/math] with vector space basis [math]\displaystyle{ \left\lbrace \begin{bmatrix}1 \\-\frac{4}{3} \\-4 \end{bmatrix} \right\rbrace }[/math] and vector space dimension 1.
  • ...
• Counter-Examples:
  • The set [math]\displaystyle{ V=\{ (x, y, z)^T \in \mathbb{R}^3|x\geqslant 0, y=-4z\} }[/math], which violates vector space closure axioms.
  • 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], which violates vector space scalar multiplication axioms.
  • Metric Spaces without vector addition, which lack vector space algebraic structure.
  • Tuple Spaces, which lack vector space scalar multiplication.
  • Affine Spaces, which lack vector space origin point.
• See: n-Dimensional Euclidean Space, Normed Vector Space, Inner Product Space, Linear Algebra Subject Area, Functional Analysis Subject Area, Metric Space, Affine Space, Banach Space, Hilbert Space, Topological Space, Linear Transformation Operation, Vector Dimension, Hyperplane, State Space, Vector Data Structure, 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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