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 $E_{\lambda }$ of equation $Y^2=X(X-1)(X-\lambda )$, we prove that, given $n$ linearly independent points $P_1(\lambda), …,P_n(\lambda)$on$E_\lambda$withcoordinatesin$\bar{$\mathbb{Q}$(\lambda)}$,thereareatmostfinitelymanycomplexnumbers$\lambda_0$suchthatthepoints$P_1(\lambda_0), …,P_n(\lambda_0)$ satisfy two independent relations on $E_{{\lambda }_0}$. This is a special case of conjectures about Unlikely Intersections on families of abelian varieties.