# Why the orthogonal complement of 0 is V?

Mathematics Asked by Ivan Bravo on December 6, 2020

I was doing an exercise and I saw this property, I would like to know why it’s true.

Let $$V$$ be a $$mathbb{K}$$-dimensional vector space and Let $$f$$ be a bilinear form. Given $$S subseteq V$$, we define: $$S^perp$$={$$alpha in V | f(alpha, beta)=0$$
$$forall beta in S$$}

{0}$$^perp$$=$$V$$

• By definition, $$S^perp$$ is subset of $$V$$, so $${0}^perp subseteq V$$.
• On the other hand, if $$alpha in V$$ is arbitrary, as $$v mapsto f(alpha,v)$$ is linear, we have that $$f(alpha,0) = 0$$, meaning that $$f(alpha,beta) = 0$$ for every $$beta in {0}$$. Thus $$alpha in {0}^perp$$ and this shows $$V subseteq {0}^perp$$.

Answered by azif00 on December 6, 2020

Every $$xin V$$ satisfies that $$f(x,0)=0$$ for $$f$$ is bilinear. And given that $${0}$$ is only composed by $$0$$ (obviously) then every $$xin V$$ is in the ortogonal complement of $$0$$.

Answered by Iesus Dave Sanz on December 6, 2020