Primordial element (algebra)

In algebra, a primordial element is a particular kind of a vector in a vector space.

Let ${\displaystyle V}$be a vector space over a field ${\displaystyle \mathbb {F} }$and let ${\displaystyle \left(e_{i}\right)_{i\in I}}$be an ${\displaystyle I}$-indexed basis of vectors for ${\displaystyle V.}$By the definition of a basis, every vector ${\displaystyle v\in V}$can be expressed uniquely as $${\displaystyle v=\sum _{i\in I}a_{i}(v)e_{i}}$$for some ${\displaystyle I}$-indexed family of scalars ${\displaystyle \left(a_{i}\right)_{i\in I}}$where all but finitely many ${\displaystyle a_{i}}$are zero. Let $${\displaystyle I(v)=\left\{i\in I:a_{i}(v)\neq 0\right\}}$$denote the set of all indices for which the expression of ${\displaystyle v}$has a nonzero coefficient. Given a subspace ${\displaystyle W}$of ${\displaystyle V,}$a nonzero vector ${\displaystyle p\in W}$is said to be primordial if it has both of the following two properties:

1. ${\displaystyle I(p)}$is minimal among the sets ${\displaystyle I(w),}$where ${\displaystyle 0\neq w\in W,}$and
2. ${\displaystyle a_{i}(p)=1}$for some index ${\displaystyle i.}$