Question

Does there exists a vector space V and a bilinear form w on V \(\oplus\)V such that w is not identically zero, but w (x,x) =0 for every x in V?
Please, help

Answer 1

@wio

Answer 2

V = {M_2 = span {\(\left[\begin{matrix}1&0\\0&0\end{matrix}\right],\left[\begin{matrix}0&1\\0&0\end{matrix}\right],\left[\begin{matrix}0&0\\1&0\end{matrix}\right],\left[\begin{matrix}0&0\\0&1\end{matrix}\right]\)}}
w \(\in V\oplus V~~ and ~~~~w (x,x) = x_1y_2 + x_2y_1\) , then w (x,x) =0 for all x
My question is : is there any other vector space else?

Answer 3

maybe something like \(w(x,y) = x-y\)? But it has to be bilinear

Answer 4

yes, I think so, but (x, x,) not (x, y)
I think of w (x,x) = x cross x , but not sure how to put it in the logic of bilinear form

Answer 5

Can I do :
in \(R^2\) w (x,x) = x1y2 - x2 y1=0 for any x,
let check. x =(x1, x2)
so, w ( x, x ) = \(x_1x_2 - x_2x_1 \) =0 for all x
does it work?

Answer 6

oh, I have to go. Thanks for being here. :)