Primordial element (algebra)

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

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