The notion of a µ-basis was developed several years ago in the context of curves implicitization [2]. From a geometric point of view, a µ-basis of a rational curve in R n is a set of n rational curves of smaller degree, that can replace the original curve for several operations like implicitizing, detecting properness, inverting, etc [2, 4, 5]. On the other hand, projective equivalences between rational curves in R n have been studied in recent years [1, 3]. The algorithms for checking projective equivalence depend heavily on the degrees of the curves to be analyzed. In this talk, we will show how projective equivalences between rational curves in R n are transferred to the elements of smallest degree of the µ-bases of the curves. These elements of smallest degree can be found without computing the whole µ-basis. As a result, we have a way to reduce the cost of computing the projective equivalences between rational curves in R n by replacing the given curves for the curves represented by the elements of smallest degree of the µ-bases of the curves, which have a much smaller degree compared to the original degree of the curves.