Let be an -dimensional vector space over a field and let be a linear transformation. Suppose is a subspace of dimension which is stable under . That is, .
- Prove that there is a basis for such that , where has dimension and has dimension .
- Prove that if there is a subspace , also invariant under , such that , then there is a basis for such that , where has dimension and has dimension .
- Prove conversely that if there is a basis of such that is block diagonal as in part (2), then there exist -invariant subspaces and such that .
We discussed block matrices in more depth in this previous exercise.
- Let be an ordered basis. Since is linearly independent, we may extend to a basis of . Since , if we write using this basis, we have and the are all 0. Thus we have .
- Let and be ordered bases. We claim that is a basis for . Certainly it is a generating set. Now suppose . Since is the direct sum of and , we have and . Thus and , and so is linearly independent. Note that, as in part (1), with respect to this basis we have , and . Likewise with . Thus we have .
- Let be an ordered basis such that is block diagonal; say , where has dimension and has dimension . Let and . Let and . Certainly and are both stable under , and .