Let $X$ be a closed subspace of a Banach space Y. I have functionals $f_0, f_1, \ldots, f_n\in X^*$ such that $f_0$ is in the span of the remaining ones. I fix an extension of $f_0$ to $Y$; let me call it $F_0$. Can I extend them to functionals $F_1, \ldots, F_n$ on $Y$ in a way that $F_0$ is in the span of $F_1, \ldots, F_n$? I don't really care about preserving the norms of the original functionals.