Write $V$ for the space of $3 \times 3$ skew-symmetric real matrices. (A) Show that for $A \in SO_3(\mathbb{R})$ and $M \in V$, $AMA^t \in V$. Write $A \cdot M$ for this action. (B) Let $\Phi : \mathbb{R}^3 \longrightarrow V$ be the map $$\begin{bmatrix} u \\ v \\ w \end{bmatrix} \mapsto \begin{bmatrix} 0 & w & -v \\ -w & 0 & u \\ v & -u & 0 \end{bmatrix}$$ With the usual action of $SO_3(\mathbb{R})$ on $\mathbb{R}^3$ and the above action on $V$, show that $\Phi(Av) = A \cdot \Phi(v)$ for every $A \in SO_3(\mathbb{R})$ and $v \in \mathbb{R}^3$. (C) Show that there does not exist $M \in V$, $M \neq 0$ such that for every $A \in SO_3(\mathbb{R})$, $A \cdot M$ belongs to the span of $M$.
Write $V$ for the space of $3 \times 3$ skew-symmetric real matrices.\\
(A) Show that for $A \in SO_3(\mathbb{R})$ and $M \in V$, $AMA^t \in V$. Write $A \cdot M$ for this action.\\
(B) Let $\Phi : \mathbb{R}^3 \longrightarrow V$ be the map
$$\begin{bmatrix} u \\ v \\ w \end{bmatrix} \mapsto \begin{bmatrix} 0 & w & -v \\ -w & 0 & u \\ v & -u & 0 \end{bmatrix}$$
With the usual action of $SO_3(\mathbb{R})$ on $\mathbb{R}^3$ and the above action on $V$, show that $\Phi(Av) = A \cdot \Phi(v)$ for every $A \in SO_3(\mathbb{R})$ and $v \in \mathbb{R}^3$.\\
(C) Show that there does not exist $M \in V$, $M \neq 0$ such that for every $A \in SO_3(\mathbb{R})$, $A \cdot M$ belongs to the span of $M$.