Past exam of the options II course of the University of Oxford / 2009 / O / O4 / i / Solution

Unbiasedness of an estimator means unbiasedness for every parameter value, not just one chosen $\theta$. Thus $DX\theta =\theta$ for every $\theta \in {\mathbb{R}}^p$. Evaluating this identity on a basis, or identifying the two linear maps, gives