Quantifier elimination for infinite underlying vector spaces

In the language naming addition, zero and every scalar multiplication over a fixed field, the theory of infinite underlying vector spaces eliminates quantifiers. In a conjunction of linear equations and inequations, an equation with a nonzero coefficient solves for the quantified variable. Otherwise the variable is excluded from only finitely many values, so infinitude supplies a witness. Disjunction and induction finish elimination. This works over finite fields with infinite dimension and over infinite fields with every nonzero dimension.