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

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

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