Approximation of multivariate functions is a difficult task when the number of input parameters is large. Identifying the directions where the function doesn't significantly vary is a key step for complexity reduction. Among other dimension reduction techniques, the Active Subspace method uses gradients of a scalar-valued function to reduce the parameter space. In this talk, we extend this methodology for vector-valued functions, e.g. functions with multiple scalar outputs or functions taking values in function spaces. Numerical examples reveal the importance of the choice of the metric to measure errors.