Restrictions as stabilizers

Two GF(q)-representations of a matroid are projectively equivalent if one can be obtained from the other by elementary row operations or column scaling. If, in addition, we allow field automorphisms, then the representations are equivalent. A matroid stabilizes its single-element extensions over GF(q), if no two GF(q)-representations of the matroid can be extended to two projectively inequivalent GF(q)-representations of its extension. We prove that if a 3-connected GF(q)-representable matroid stabilizes its single-element extensions, then it stabilizes any GF(q)-representable matroid that contains it as a restriction. As a result, a GF(q)-representable matroid with a line containing at least q points is stabilized by that line and a GF(q)-representable matroid with a plane containing at least 2q points, but no line containing at least q points, is stabilized by that plane.