 Independence of a pair is not definable over an infinite field

In the one-sorted vector-space language naming each scalar multiplication, a first-order formula reduces by quantifier elimination to finitely many linear equations. An independent pair satisfies only the identically zero equations among them. A dependent pair $(x,ax)$ with nonzero $x$ satisfies the same listed equations if the scalar $a$ avoids their finitely many forbidden ratios, which is possible over an infinite field. Thus no formula defines pair independence. The complete independent-pair type is still consistent in an infinite-dimensional model; it is nonisolated.