For a singular linear model ${\cal A}= (y ,{X \beta},$ {\si{2}} $V)$ and its transformed model ${\cal A_{F}}=(Fy,FX\beta, \sigma^2FV F')$, where V is nonnegative definite and $X$ can be rank-deficient, the expressions for the
differences of the estimates for the vector of $FX\beta$ and the variance factor {\si{2}} are given. Moreover, the necessary and sufficient conditions for the equalities of the estimates for the vector of $FX\beta$ and the variance factor {\si{2}} are also established. In the meantime, works in Baksalary and Kala (1981) are strengthened and consequences in Puntanen and Nurhonen (1992), and Puntanen (1996) are extended.