# 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 $(1,2,\mathrm{\dots},n)$ to $\mathbb{K}$. Such a mapping $a\in {\mathbb{K}}^{n}$ is just a formal way of speaking of a list of field elements ${a}^{1},\mathrm{\dots},{a}^{n}\in \mathbb{K}$.

The above description is somewhat restrictive. A more flexible
definition of a list vector is the following. Let $I$ be a finite
list of indices^{1}^{1}Distinct index sets^{} are often used when
working with multiple frames of reference., $I=(1,\mathrm{\dots},n)$ 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

${(u+v)}^{i}$ | $={u}^{i}+{v}^{i},i\in I,$ | ||

${(ku)}^{i}$ | $=k{u}^{i},i\in I.$ |

We also have the zero vector^{} $\mathrm{\U0001d7ce}\in {\mathbb{K}}^{I}$, namely the constant mapping

$${\mathrm{\U0001d7ce}}^{i}=0,i\in I.$$ |

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

$$a=\left(\begin{array}{c}\hfill {a}^{1}\hfill \\ \hfill {a}^{2}\hfill \\ \hfill \mathrm{\vdots}\hfill \\ \hfill {a}^{n}\hfill \end{array}\right).$$ |

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

$$\alpha =({\alpha}_{1},\mathrm{\dots},{\alpha}_{n}).$$ |

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 |