Extra Credits 1 (due March 22)

Let $\mathbb R ^+:= \{x \in \mathbb R, x> 0\}$. For any $u , v \in \mathbb R^+$, $c \in \mathbb R$, define $u \oplus v = uv$ and $c \odot u = u ^c$. Prove that $(\mathbb R^+, \oplus, \odot)$ is a real vector space.

Extra Credits 2 (due April 27th)

Find all matrices which are symmetric and orthogonal and explain why.