Linear combinations of an MVUE are not MVUE

Let $T_0$ be an MVU estimator of $\gamma(\theta)$ and $T_1$ an unbiased estimator with efficiency $e_\theta<1$. Which of the following statements is/are correct?

I. The unbiased linear combination of $T_0$ and $T_1$, i.e., $U=a_0T_0+a_1T_1$; $a_1\ne0$, $a_0+a_1=1$ is MVU.

II. The unbiased linear combination of $T_0$ and $T_1$, i.e., $V=a_0T_0-a_1T_1$; $a_1\ne0$, $a_0-a_1=1$ is MVU.