Linear relations in families of powers of elliptic curves

Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve Eλ of equation Y2 = X (X − 1)(X − λ), we prove that, given n linearly independent points P1(λ),…, Pn (λ) on Eλ with coordinates in Q(λ), there are at most finitely many complex numbers λ0 such that the points P1(λ0),…, Pn (λ0) satisfy two independent relations on Eλ0. This is a special case of conjectures about unlikely intersections on families of abelian varieties.