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Homelist vector

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# 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 indices^{1}^{1}Distinct 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

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

## Mathematics Subject Classification

15A03*no label found*15A90

*no label found*

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