= Solution
Use the vector-space <axioms>: addition is an <abelian group> operation with identity zero; $\lambda_1(x)=x$; $\lambda_a\lambda_b(x)=\lambda_{ab}(x)$; $\lambda_a(x+y)=\lambda_a(x)+\lambda_a(y)$; and $\lambda_{a+b}(x)=\lambda_a(x)+\lambda_b(x)$, with $\lambda_0(x)=0$. <Group> inverses may be expressed by $\lambda_{-1}$, so no extra inverse symbol is needed.
Add the dimension-unbounded scheme
$$
\boxed{D_n:\exists x_1\cdots\exists x_n\operatorname{LI}_n(\bar x)\qquad(n\ge1).}
$$
Over a <finite field>, a finite-dimensional space has finitely many <Elements>, while every infinite-dimensional space is infinite. Hence this scheme axiomatizes exactly the infinite underlying <vector spaces> requested. In this finite-field case the vector-space <axiom> list itself is finite; the infinitude of the additional <dimension of a vector space>[dimension] scheme is substantive.
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