(Solved): 4. Determine if the following give examples of subspaces. If the given \( S \) is a subspace, prov ...

4. Determine if the following give examples of subspaces. If the given \( S \) is a subspace, prove it. If not, then prove that it isn't a subspace. (a) \( S=\left\{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}: x_{1}-3 x_{3}=x_{2}\right\} \) in \( \mathbb{R}^{3} \). (b) \( S=\{f \in C[a, b]: f(0)+f(1)=0\} \) in the vector space \( C[a, b] \) (where \( a, b \) are real numbers and \( a