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


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


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

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