Skew-symmetric matrices and invariants

Question

Consider the set of skew-symmetric matrices $AS(n)=\{M \in M(n,n,\mathbb{R}): M^T=-M\}$.

How to prove that, to the inner dot product $\left\langle .,. \right\rangle$ of $\mathbb{R^n}$, if $W$ is invariant for $M \in AS(n)$, then $W^{\bot}$ is invariant?

Thank you

Answer 1

Let $w$ $\in$ $W^{\perp} $ $\Rightarrow$ $\langle v,w\rangle = 0$, $\forall$ $v$ $\in$ $W$.

Now, note that $\forall$ $v$ $\in$ $W$

$$\langle v , M w \rangle = \langle M^T v , w \rangle = - \langle M v , w \rangle = 0 $$

because $M v$ $\in$ $W$ $\Rightarrow$ $M w$ $\in$ $W^{\perp}$, $\forall$ $w$ $\in W^{\perp}$.

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If the above computation wasn't clear

$$\langle v , M w \rangle = v^T\cdot M w = - v^T \cdot M^T w = -(M v)^T \cdot w = -\langle M v , w \rangle $$