Why the orthogonal complement of 0 is V?

Question

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}$^\perp$=$V$

Answer 1

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

Answer 2

• 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 $V \subseteq \{0\}^\perp$.