[ \star \star \star ] Let V=\mathbb{M}_3(\mathbb{C}), the set of all 3 \times 3 matrices with entries in \mathbb{C}. Clearly V is a vector space over \mathbb{C} and \dim_{\mathbb{C}} V= 9.

Let W be a vector subspace of V such that uv \in W for all u,v \in W. Prove that \dim_{\mathbb{C}} W \neq 8.