Linear Equations in Singular Moduli

We establish an effective version of the Andr'e-Oort conjecture for linear subspaces of $Y(1{)}_{\mathbb{C}}^n\approx {\mathbb{A}}_{\mathbb{C}}^n$. Apart from the trivial examples provided by weakly special subvarieties, this yields the first algebraic subvarieties in a Shimura variety of dimension $>1$ whose CM-points can be (theoretically) determined.