Primordial element (algebra)

In algebra, a primordial element is a particular kind of a vector in a vector space. Let V be a vector space over a field k and fix a basis for V of vectors e i for i ∈ I . By the definition of a basis, every vector v in V can be expressed uniquely as

Define I ( v ) = { i ∈ I ∣ a i ( v ) ≠ 0 } , the set of indices for which the expression of v has a nonzero coefficient. Given a subspace W of V, a nonzero vector w in W is said to be "primordial" if it has the following two properties:

1. I ( w ) is minimal among the sets I ( w ′ ) , 0 ≠ w ′ ∈ W and
2. a i ( w ) = 1 for some i