In a linear (or affine) functional model the principal parameter is a subspace (respectively an affine subspace) in a finite dimensional inner product space, which contains the means of $n$ multivariate normal populations, all having the same covariance matrix. A relatively simple, essentially algebraic derivation of the maximum likelihood estimates is given, when these estimates are based on single observed vectors from each of the $n$ populations and an independent estimate of the common covariance matrix. A new derivation of least squares estimates is also given.
C. Villegas. "Maximum Likelihood and Least Squares Estimation in Linear and Affine Functional Models." Ann. Statist. 10 (1) 256 - 265, March, 1982. https://doi.org/10.1214/aos/1176345708