Let $B \in M_3(\mathbb{R})$ be antisymmetric. a) Show that $\det B = 0$. b) Show that $\left(\operatorname{Ker} u_B\right)^\perp$ is stable under $u_B$. c) Deduce that $B$ has rank 0 or 2.
Let $B \in M_3(\mathbb{R})$ be antisymmetric.
a) Show that $\det B = 0$.
b) Show that $\left(\operatorname{Ker} u_B\right)^\perp$ is stable under $u_B$.
c) Deduce that $B$ has rank 0 or 2.