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