PlanetMath: list vector

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (38)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

list vector

(Definition)

Let $\mathbb{K}$ be a field and a positive natural number. We define $\mathbb{K}^n$ to be the set of all mappings from the index list $(1,2,\ldots,n)$ to $\mathbb{K}$ . Such a mapping $a\in \mathbb{K}^n$ is just a formal way of speaking of a list of field elements $a^1,\ldots, a^n\in\mathbb{K}$ .

The above description is somewhat restrictive. A more flexible definition of a list vector is the following. Let be a finite list of indices ¹, $I=(1,\ldots,n)$ is one such possibility, and let $\mathbb{K}^I$ denote the set of all mappings from to $\mathbb{K}$ . A list vector, an element of $\mathbb{K}^I$ , is just such a mapping. Conventionally, superscripts are used to denote the values of a list vector, i.e. for $u\in \mathbb{K}^I$ and $i\in I$ , we write instead of .

We add and scale list vectors point-wise, i.e. for $u, v\in \mathbb{K}^I$ and $k\in \mathbb{K}$ , we define $u+v\in \mathbb{K}^I$ and $ku\in \mathbb{K}^I$ , respectively by

$\displaystyle (u+v)^i$	$\displaystyle = u^i+v^i,\quad i\in I,$
$\displaystyle (ku)^i$	$\displaystyle = k u^i,\quad i\in I.$

We also have the zero vector $\mathbf{0}\in \mathbb{K}^I$ , namely the constant mapping

$\displaystyle \mathbf{0}^i = 0,\quad i\in I.$

The above operations give $\mathbb{K}^I$ the structure of an (abstract) vector space over $\mathbb{K}$ .

Long-standing traditions of linear algebra hold that elements of $\mathbb{K}^I$ be regarded as column vectors. For example, we write $a\in \mathbb{K}^n$ as

$\displaystyle a= \begin{pmatrix} a^1 \\ a^2 \\ \vdots \\ a^n \end{pmatrix}.$

Row vectors are usually taken to represents linear forms on $\mathbb{K}^I$ . In other words, row vectors are elements of the dual space $\left(\mathbb{K}^I\right)^*$ . The components of a row vector are customarily written with subscripts, rather than superscripts. Thus, we express a row vector $\alpha\in\left(\mathbb{K}^n\right)^*$ as

$\displaystyle \alpha = (\alpha_1,\ldots,\alpha_n).$

Footnotes

...¹: Distinct index sets are often used when working with multiple frames of reference.

"list vector" is owned by rmilson.

(view preamble)

Also defines:

column vector, row vector

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: subscripts, components, dual space, linear forms, represents, linear algebra, vector space, structure, operations, constant mapping, zero vector, superscripts, frames, multiple, index sets, indices, finite, index, mappings, natural number, positive, field
There are 47 references to this entry.

This is version 2 of list vector, born on 2002-07-24, modified 2002-10-19.
Object id is 3200, canonical name is ListVector.
Accessed 10259 times total.

Classification:

AMS MSC:	15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)
	15A90 (Linear and multilinear algebra; matrix theory :: Applications of matrix theory to physics)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact