Vector Space Point

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





  • (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.


  • (Wikipedia, 2011) ⇒
    • 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. ...


  • (WordNet, 2009) ⇒
    • 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)



    • 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.