Assume that a linear random-effects model (LRM) $\by = \bX \bbe + \bve = \bX\bbe+ \bve$ with $\bbe = \bA \bal + \bga$ is transformed as $\bT\by = \bT\bX\bbe + \bT\bve = \bT\bX\bA \bal + \bT\bX\bga + \bT\bve$ by pre-multiplying a given matrix $\bT$. Estimations/predictions of the unknown parameters under the two models are not necessarily the same because the transformation matrix $\bT$ occurs in the statistical inference of the transformed model. This paper presents a general algebraic approach to the problem of best linear unbiased prediction (BLUP) of a joint vector all unknown parameters in the LRM and its linear transformations, and provides a group of fundamental and comprehensive results on mathematical and statistical properties of the BLUP under the LRM.