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 ¹, $I=(1,\ldots,n)$ 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

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

Footnotes

...http://planetmath.org/encyclopedia/IndexedBy.html ¹

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