I'm having trouble with subsets and subspaces (proper way of showing whether it's a subspace or not) I usually go over the 3 statements that have to be true for it to be a subspace, which is : 1. There exists a 0 vector 2. U and V such that U + V are in the subspace 3. KU is also in S. I'd really appreciate if somebody would go over the following questions with a step by step explanation.
1. is the vector space of all real-valued functions defined on the interval , and is the subset of consisting of those functions satisfying
2. , and is the subset of consisting of those functions satisfying the differential equation
3. , and is the subset of all upper triangular matrices.