list vector (original) (raw)

Let 𝕂 be a field and n a positive natural number. We define𝕂n to be the set of all mappings from the index list(1,2,…,n) to 𝕂. Such a mapping a∈𝕂n is just a formal way of speaking of a list of field elements a1,…,an∈𝕂.

The above description is somewhat restrictive. A more flexibledefinition of a list vector is the following. Let I be a finite list of indices11Distinct index sets are often used when working with multiple frames of reference., I=(1,…,n) is one such possibility, and let 𝕂I denote the set of all mappings from I to 𝕂. A list vector, an element of 𝕂I, is just such a mapping. Conventionally, superscripts are used to denote the values of a list vector, i.e. for u∈𝕂I and i∈I, we write ui instead of u⁢(i).

We add and scale list vectors point-wise, i.e. for u,v∈𝕂I and k∈𝕂, we define u+v∈𝕂I and k⁢u∈𝕂I, respectively by

We also have the zero vector 𝟎∈𝕂I, namely the constant mapping

The above operations give 𝕂I thestructure of an (abstract) vector space over 𝕂.