Linear independence of values of $E$-functions

We prove a general theorem that establishes a relation between linear and algebraic independence of values at algebraic points of $E$-functions and properties of the ideal formed by all algebraic equations relating these functions over the field of rational functions. Using this theorem we prove sufficient conditions for linear independence of values of $E$-functions as well as for algebraic independence of values of subjects of them. The main result is an assertion stating that at all algebraic points, except finitely many, the values of $E$-functions are linearly independent over the field of all algebraic numbers if the corresponding functions are linearly independent over the field of rational functions. The theorem is applied to concrete $E$-functions.