The concept of differential flatness is an elegant approach to system analysis and the design of controllers and observers. In many cases, however, the main difficulty is the determination of a flat output. For holonomic mechanical systems the notion of 'configuration flatness' was introduced some years ago, meaning that a flat output is a function only of configuration variables (and not of the velocities nor the input). Obviously, such a property could drastically facilitate the search for a flat output. In this contribution, we show on the basis of a polynomial matrix representation that for linear conservative mechanical systems left coprimeness (i. e., controllability), which is equivalent to (general) flatness, implies configuration flatness.