Let $V$ be a subspace of the complex vector space $M_n(\mathbb{C})$. Suppose that every non-zero element of $V$ is an invertible matrix. Show that $\dim_{\mathbb{C}} V \leq 1$.
Let $V$ be a subspace of the complex vector space $M_n(\mathbb{C})$. Suppose that every non-zero element of $V$ is an invertible matrix. Show that $\dim_{\mathbb{C}} V \leq 1$.