This paper is concerned with the relationships between best linear minimum biased predictors (BLMBPs) in the context of a general linear model (GLM) and its transformed general linear models (TGLMs). We shall establish a mathematical procedure by means of some exact and analytical tools in matrix theory that were developed in recent years. The coverage includes constructing a general vector composed of all unknown parameters in the context of a GLM and its TGLMs, deriving the exact expressions of the BLMBPs through the technical use of analytical solutions of a constrained quadratic matrix-valued function optimization problem in the Löwner partial ordering, and discussing a variety of theoretical performances and properties of the BLMBPs. We also give a series of characterizations of relationships between BLMBPs under a given GLM and its TGLMs.