Vector Space Point

A vector space point is a numeric tuple (with vector members of vector dimensions) that is a member of in some vector space.

• AKA: One Dimensional Array.
• Context:
  • It can range from being an Abstract Vector to being a Vector Record.
  • It can range from being a Scalar Vector, to being an Integer Vector, to being a Boolean Vector.
  • It can range from being a Low-Dimensional Vector (e.g. a 2d vector) to being a High-Dimensional Vector.
  • It can be a member of a Vector Set or Vector List (such as a matrix).
  • It can (often) be a Task Input to a Vector Operation (such as vector length).
  • It can (often)be a Task Output of a Vector-Valued Operation.
• Example(s):
  • [math]\displaystyle{ \vec{x} = (0.9, 0.7, 0.4) }[/math].
  • [math]\displaystyle{ \vec{v} = (111.2, 41.0, 35.4, 22.0, 33.4) }[/math].
  • [math]\displaystyle{ \vec{y} = (1, 4, 2, 7) }[/math], an integer vector.
  • a Document Vector.
  • a Zero Vector.
  • an Identity Vector.
  • …
• Counter-Example(s):
  • a Tuple, such as [math]\displaystyle{ (0.9, RED, 3.4, 0, 1) }[/math] or a Binary Tuple, such as: (0,1,1,0,0,1,0,0,1).
  • a Scalar, such as [math]\displaystyle{ 3.1 }[/math]
• See: Vector Operation, Array, Vectorized Representation, Numerical Attribute, Vector Field.

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References

2015a

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/vector_(mathematics_and_physics) Retrieved:2015-3-2.
  • When used without any further description, vector refers either to:
    • Most generally, an element of a vector space.
    • In physics and geometry, an Euclidean vector, used to represent physical quantities that have both magnitude and direction
  • Vector can also have a variety of different meanings depending on context.

2015b

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/vector_(mathematics_and_physics)#Vectors Retrieved:2015-3-2.
  • An element of a vector space
    • An element of the real coordinate space 'Rn
    • Basis vector, one of a set of vectors (a "basis") that, in linear combination, can represent every vector in a given vector space
    • Column vector or row vector, a one-dimensional matrix often representing the solution of a system of linear equations
    • Coordinate vector, in linear algebra, an explicit representation of an element of any abstract vector space
  • Axial vector, or pseudovector, a quantity that transforms like a vector under proper rotation but not generally under reflection
  • Darboux vector, the areal velocity vector of the Frenet frame of a space curve
  • Displacement vector, a vector that specifies the change in position of a point relative to a previous position
  • Euclidean vector, a geometric entity endowed with magnitude and direction as well as a positive-definite inner product; an element of a Euclidean vector space. In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, such as force, in contrast to scalar quantities, which have no direction.
    • Burgers vector, a vector that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice
    • Laplace–Runge–Lenz vector, a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another
    • Normal vector, or surface normal, a vector that is perpendicular to a (hyper)surface at a point
    • Vector product, or cross product, an operation on two vectors in a three-dimensional Euclidean space, producing a third three-dimensional Euclidean vector
  • Four-vector, in the theory of relativity, a vector in a four-dimensional real vector space called Minkowski space
  • Gradient vector, the vector giving the magnitude and direction of maximum increase of a scalar field.
  • Gyrovector, a hyperbolic geometry version of a vector
  • Interval vector, in musical set theory, an array that expresses the intervallic content of a pitch-class set
  • Null vector, a vector whose magnitude is zero
  • P-vector, the tensor obtained by taking linear combinations of the wedge product of p tangent vectors
  • Position vector, a vector representing the position of a point in an affine space in relation to a reference point
  • Poynting vector, in physics, a vector representing the energy flux density of an electromagnetic field.
  • Probability vector, in statistics, a vector with non-negative entries that sum to one
  • Random vector or multivariate random variable, in statistics, a set of real-valued random variables that may be correlated.
  • Spin vector, or Spinor, is an element of a complex vector space introduced to expand the notion of spatial vector
  • Tangent vector, an element of the tangent space of a curve, a surface or, more generally, a differential manifold at a given point.
  • The vector part of a quaternion, a mathematical entity that is one possible generalisation of a vector
  • Tuple, an ordered list of numbers, sometimes used to represent a vector
  • Unit vector, a vector in a normed vector space whose length is 1
  • Wave vector, a vector representation of the local phase evolution of a wave

2012

• (Golub & Van Loan, 2012) ⇒ Gene H. Golub, and Charles F. Van Loan. (2012). “Matrix Computations (4th Ed.)." Johns Hopkins University Press. ISBN:1421408597
  • QUOTE: Let [math]\displaystyle{ \R^n }[/math] denote the vector space of real n-vectors: :[math]\displaystyle{ x \in \mathbb{R}^n \Leftrightarrow \ \ x = \begin{bmatrix} x_{1} \\ \vdots \\ x_n \end{bmatrix} \ \ x_i \in \R. }[/math] We refer to [math]\displaystyle{ x_i }[/math] as the ith component of [math]\displaystyle{ x }[/math]. Depending upon context, the alternative notations [math]\displaystyle{ [x]_i }[/math] and [math]\displaystyle{ x(i) }[/math] are sometimes used.

2011

• (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/Vector_(mathematics_and_physics)
  • QUOTE: In mathematics and physics, a vector is an element of a vector space. If n is a non negative integer and K is either the field of the real numbers or the field of the complex number, then [math]\displaystyle{ K^n }[/math] is naturally endowed with a structure of vector space, where [math]\displaystyle{ K^n }[/math] is the set of the ordered sequences of n elements of K. It follows that, in many cases, vector simply refers to a sequence of fixed length of real or complex numbers. The various uses of vector which follow are special instances of this general definition.

     Euclidean vector, a geometric entity endowed with both length and direction; an element of a Euclidean vector space. In physics, euclidean vectors are used to represent physical quantities which have both magnitude and direction, such as force, in contrast to scalar quantities, which have no direction. ...

2009a

• (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=vector
  • S: (n) vector (a variable quantity that can be resolved into components)
  • S: (n) vector (a straight line segment whose length is magnitude and whose orientation in space is direction)
  • S: (n) vector, transmitter (any agent (person or animal or microorganism) that carries and transmits a disease) "mosquitos are vectors of malaria and yellow fever"; "fleas are vectors of the plague"; "aphids are transmitters of plant diseases"; "when medical scientists talk about vectors they are usually talking about insects"
  • S: (n) vector ((genetics) a virus or other agent that is used to deliver DNA to a cell)

2009b

• (Math Website, 2009) ⇒ http://www.math.com/tables/oddsends/vectordefs.htm
  • Definition: A vector of dimension n is an ordered collection of n elements, which are called components. ... It can represent magnitude and direction simultaneously.

1999

• http://www.vias.org/tmdatanaleng/cc_vector_intro.html
  • QUITE: We define an ordered set of n equal objects written in a column vector of order n, and the row counterpart of m objects a row vector (of order m). Please keep in mind that these definitions are simplified and cover only part of the exact, mathematical definition. However, the definition given here is sufficient for our purposes concerning data analysis.

    To denote a specific vector, we shall use a lowercase, bold letter, such as a, for example. Whether this vector a is a column or row vector, will usually be clear from the context in which the letter is used. When written explicitly, vectors are put in parenthesis.

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