Primordial element (algebra)

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

Definition

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

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