list vector
Let be a field and a positive natural number. We define to be the set of all mappings from the index list to . Such a mapping is just a formal way of speaking of a list of field elements .
The above description is somewhat restrictive. A more flexible definition of a list vector is the following. Let be a finite list of indices11Distinct index sets are often used when working with multiple frames of reference., is one such possibility, and let denote the set of all mappings from to . A list vector, an element of , is just such a mapping. Conventionally, superscripts are used to denote the values of a list vector, i.e. for and , we write instead of .
We add and scale list vectors point-wise, i.e. for and , we define and , respectively by
We also have the zero vector , namely the constant mapping
The above operations give the structure of an (abstract) vector space over .
Long-standing traditions of linear algebra hold that elements of be regarded as column vectors. For example, we write as
Row vectors are usually taken to represents linear forms on . In other words, row vectors are elements of the dual space . The components of a row vector are customarily written with subscripts, rather than superscripts. Thus, we express a row vector as
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 |