list vector


Let 𝕂 be a field and n a positive natural number. We define 𝕂n to be the set of all mappings from the index list (1,2,,n) to 𝕂. Such a mapping a𝕂n is just a formal way of speaking of a list of field elements a1,,an𝕂.

The above description is somewhat restrictive. A more flexible definition of a list vector is the following. Let I be a finite list of indices11Distinct index setsMathworldPlanetmathPlanetmath are often used when working with multiple frames of reference., I=(1,,n) is one such possibility, and let 𝕂I denote the set of all mappings from I to 𝕂. A list vector, an element of 𝕂I, is just such a mapping. Conventionally, superscripts are used to denote the values of a list vector, i.e. for u𝕂I and iI, we write ui instead of u(i).

We add and scale list vectors point-wise, i.e. for u,v𝕂I and k𝕂, we define u+v𝕂I and ku𝕂I, respectively by

(u+v)i =ui+vi,iI,
(ku)i =kui,iI.

We also have the zero vectorMathworldPlanetmath 𝟎𝕂I, namely the constant mapping

𝟎i=0,iI.

The above operationsMathworldPlanetmath give 𝕂I the structureMathworldPlanetmath of an (abstract) vector space over 𝕂.

Long-standing traditions of linear algebra hold that elements of 𝕂I be regarded as column vectors. For example, we write a𝕂n as

a=(a1a2an).

Row vectors are usually taken to represents linear forms on 𝕂I. In other words, row vectors are elements of the dual spacePlanetmathPlanetmath (𝕂I)*. The componentsMathworldPlanetmathPlanetmath of a row vector are customarily written with subscripts, rather than superscripts. Thus, we express a row vector α(𝕂n)* as

α=(α1,,αn).
Title list vector
Canonical name ListVector
Date of creation 2013-03-22 12:51:50
Last modified on 2013-03-22 12:51:50
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 5
Author rmilson (146)
Entry type Definition
Classification msc 15A03
Classification msc 15A90
Defines column vector
Defines row vector