# Linear Combinations of Least Square Estimator

I come across a problem about finding the least square estimator of A$$\beta$$, where $$\beta$$ is the parameter vector in linear model ($$Y=X\beta+\epsilon$$). My question is, would the least square estimator of A$$\beta$$ be Ab, where b is the least square estimator of $$\beta$$? I know that Ab is BLUE (best linear unbiased estimator) of A$$\beta$$, but I'm not sure if this property guarantees it to be the least square estimator of A$$\beta$$. Can you guys help me prove/disprove it? Many thanks.

Assume $$A$$ to be an invertible matrix of constants, Let $$\gamma = A\beta$$ then for the model $$Y = X\beta + \epsilon = XA^{-1}\gamma + \epsilon$$. Hence $$\hat{\gamma} = ({A'}^{-1}X'XA^{-1})^{-1}A'^{-1}X'Y = A(X'X)^{-1}(A'A'^{-1})X'Y = A(X'X)^{-1}X'Y = A\hat{\beta}$$