In algebra, a primordial element is a particular kind of a vector in a vector space.
Definition Let V {\displaystyle V} be a vector space over a field F {\displaystyle \mathbb {F} } and let ( e i ) i ∈ I {\displaystyle \left(e_{i}\right)_{i\in I}} be an I {\displaystyle I} -indexed basis of vectors for V . {\displaystyle V.} By the definition of a basis, every vector v ∈ V {\displaystyle v\in V} can be expressed uniquely as
v = ∑ i ∈ I a i ( v ) e i {\displaystyle v=\sum _{i\in I}a_{i}(v)e_{i}}
for some I {\displaystyle I} -indexed family of scalars ( a i ) i ∈ I {\displaystyle \left(a_{i}\right)_{i\in I}} where all but finitely many a i {\displaystyle a_{i}} are zero. Let
I ( v ) = { i ∈ I : a i ( v ) ≠ 0 } {\displaystyle I(v)=\left\{i\in I:a_{i}(v)\neq 0\right\}}
denote the set of all indices for which the expression of v {\displaystyle v} has a nonzero coefficient. Given a subspace W {\displaystyle W} of V , {\displaystyle V,} a nonzero vector p ∈ W {\displaystyle p\in W} is said to be primordial if it has both of the following two properties:
I ( p ) {\displaystyle I(p)} is minimal among the sets I ( w ) , {\displaystyle I(w),} where 0 ≠ w ∈ W , {\displaystyle 0\neq w\in W,} and
a i ( p ) = 1 {\displaystyle a_{i}(p)=1} for some index i . {\displaystyle i.}
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