Question

For each of the sets \(\mathcal{S}\) below, determine if it is a linear subspace of the given real vector space \(V\). If it is a subspace, write down a basis for it.

a) \(V=\operatorname{Mat}_{3 \times 3}(\mathbb{R}), \mathcal{S}=\{A \in V \mid \operatorname{rank}(A)=3\}\).
b) \(V=\operatorname{Mat}_{2 \times 2}(\mathbb{R}), \mathcal{S}=\left\{\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \in V \mid a+d=0\right\}\).