Prove that if $V$ is a finite dimensional inner space and $V=U \oplus W$ then $V=U^{\perp} \oplus W^{\perp}$

Question

I've tried many different ways and each time I get lost in the way. In the previous (related) questions I have proven the following:

1. $(U+W)^{\perp} = U^{\perp} + W^{\perp}$
2. If $V$ is finite dimensional then $(U \cap W)^{\perp} = U^{\perp} + W^{\perp}$

Maybe one of those might help...

Answer 1

Assume that there is a common element $v \in U^{\bot} \cap W^{\bot}$, then we have $$v^T u = 0, v^T w =0$$ for all choices of $u \in U$ and $w \in W$ Now $v$ itself can be written as $v = u_0 + w_0$ for some $u_0 \in U$ and $w_0 \in W$. Hence: $$v^Tv =v^T(u_0+w_0)=0$$ It follows that $v=0$ is the only common element.