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!
Best Answer
Proof that the Ker of 1st definition is contained in second: If $y\in Ker(\alpha)$ then $\alpha(x,y)=0$ for each $x\in V$. It is obvious that basis vectors are in $V$ so $\alpha(e_i,y)=0$.
Proof that the Ker of 2nd definition is contained in the first: If $\alpha(e_i,y)=0$ and we take any $x\in V$ we have
$$x=\sum \lambda_i e_i$$
therefore $$\alpha(x,y)=\sum\lambda_i\alpha(e_i,y)=0$$
Remark: In fact, there is no need $\{e_i\}$ to be a basis. They suffice to be a generators system of $V$.