list vector

Let $\mathbb{K}$ be a field and $n$ 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 $I$ be a finite list of indices¹¹Distinct index sets are often used when working with multiple frames of reference., $I=(1,\ldots,n)$ is one such possibility, and let $\mathbb{K}^{I}$ denote the set of all mappings from $I$ 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 $u^{i}$ instead of $u(i)$ .

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

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

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

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

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

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

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

$\alpha=(\alpha_{1},\ldots,\alpha_{n}).$

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