For these problems:

1. Define the annihilator of as

, where is a subset of finite-dimensional vector space .

2. Let be finite-dimensional vector spaces with ordered bases respectively. Then for any linear mapping , the mapping is defined by for all with the property that .

Question 1: Suppose that is a finite-dimensional vector space and is linear. Prove that .

Question 2: Let be a linear operator on and let be a subspace of . Prove that is -invariant if and only if is -invariant.