Primordial element (algebra)

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

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

1. ${\displaystyle I(p)}$ is minimal among the sets ${\displaystyle I(w),}$ where ${\displaystyle 0\ne w\in W,}$ and
2. ${\displaystyle a_i(p)=1}$ for some index ${\displaystyle i.}$

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