The problem considered here is a multivariate generalization of the classical Behrens-Fisher problem. Based on samples of sizeni,i= 1, …,k, fromkmultivariate normal populations (of perhaps different dimension), a test statistic is proposed to test null hypotheses about linear relationships between components of the mean vectors of thekpopulations. Under the null hypothesis, the test statistic has a centralFdistribution and under alternatives, a non-centralFdistribution. For three different specializations of the above problem, the proposed test statistic reduces to tests proposed by Scheffé, Bennett, and Anderson respectively.