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 ${\displaystyle e_{i}}$ for ${\displaystyle i\in I}$. By the definition of a basis, every vector v in V can be expressed uniquely as

${\displaystyle v=\sum _{i\in I}a_{i}(v)e_{i}.}$

Define ${\displaystyle I(v)=\{i\in I\mid a_{i}(v)\neq 0\}}$, 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]

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