ListVector 2002-07-24 07:33:04 2002-10-19 09:45:26 Definition vector column vector row vector \newcommand{\cU}{\mathcal{U}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\hbF}{\hat{\bF}} \newcommand{\va}{a} \newcommand{\vu}{u} \newcommand{\vv}{v} \newcommand{\bzero}{\mathbf{0}} \newcommand{\supt}{^{\scriptscriptstyle\mathrm{T}}} \newcommand{\kfield}{\mathbb{K}} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand{\reals}{\mathbb{R}} \newcommand{\natnums}{\mathbb{N}} \newcommand{\cnums}{\mathbb{C}} \newcommand{\znums}{\mathbb{Z}} \newcommand{\lp}{\left(} \newcommand{\rp}{\right)} \newcommand{\lb}{\left[} \newcommand{\rb}{\right]} \newcommand{\supth}{^{\text{th}}} \newtheorem{proposition}{Proposition} \newtheorem{definition}[proposition]{Definition} \newcommand{\nl}[1]{{\PMlinkescapetext{{#1}}}} \newcommand{\pln}[2]{{\PMlinkname{{#1}}{#2}}} Let $\kfield$ be a field and $n$ a positive natural number. We define $\kfield^n$ to be the set of all mappings from the index list $(1,2,\ldots,n)$ to $\kfield$. Such a mapping $\va\in \kfield^n$ is just a formal way of speaking of a list of field elements $\va^1,\ldots, \va^n\in\kfield$. The above description is somewhat restrictive. A more flexible definition of a list vector is the following. Let $I$ be a finite list of indices\footnote{Distinct index sets are often used when working with multiple frames of reference.}, $I=(1,\ldots,n)$ is one such possibility, and let $\kfield^I$ denote the set of all mappings from $I$ to $\kfield$. A list vector, an element of $\kfield^I$, is just such a mapping. Conventionally, superscripts are used to denote the values of a list vector, i.e. for $\vu\in \kfield^I$ and $i\in I$, we write $\vu^i$ instead of $\vu(i)$. We add and scale list vectors point-wise, i.e. for $\vu, \vv \in \kfield^I$ and $k\in \kfield$, we define $\vu+\vv\in \kfield^I$ and $k\vu\in \kfield^I$, respectively by \begin{align*} (\vu+\vv)^i &= \vu^i+\vv^i,\quad i\in I,\\ (k\vu)^i &= k \vu^i,\quad i\in I. \end{align*} We also have the zero vector $\bzero\in \kfield^I$, namely the constant mapping $$\bzero^i = 0,\quad i\in I.$$ The above operations give $\kfield^I$ the structure of an (abstract) vector space over $\kfield$. Long-standing traditions of linear algebra hold that elements of $\kfield^I$ be regarded as column vectors. For example, we write $\va\in \kfield^n$ as $$\va = \begin{pmatrix} \va^1 \\ \va^2 \\ \vdots \\ \va^n \end{pmatrix}.$$ Row vectors are usually taken to represents linear forms on $\kfield^I$. In other words, row vectors are elements of the dual space $\lp\kfield^I\rp^*$. The components of a row vector are customarily written with subscripts, rather than superscripts. Thus, we express a row vector $\alpha\in\lp\kfield^n\rp^*$ as $$\alpha = (\alpha_1,\ldots,\alpha_n).$$