It is written in book, I read: **kernel of bilinear form is space consisting of vectors $y$, such:**

$$Ker(\alpha)=\{y\in V:\alpha(x,y)=0,\ \forall x\in V\}$$
Nice I get it, but then it is said, that **definition on top is equal to:**

$$Ker(\alpha)=\{y\in V:\alpha(e\_i,y)=0,\ i=1,...,n\},$$
where $e\_1,..e\_n$ are basis vectors. That is the part I don't understand. I think, that we can transform definition using linear properties of $\alpha$:
$$x=x\_1e\_1 + ... + x\_ne\_n\\
Ker(\alpha)=\{y\in V:x\_1\alpha(e\_1,y)+..+x\_n\alpha(e\_n,y)=0,\ i=1,...,n, \ \forall x\in V\}$$

So please, how are these definitions equivalent?
Thank you!