# Left- and Right- Kernels of Bilinear maps B:VxV->K for V.Spaces

1. Apr 21, 2011

### Bacle

Hi, all:

Let V be a vector space over k, and let B be a bilinear map into k,
i.e.:

B:VxV-->k

Define the left-kernel of B to be the set of A in V with

B (A,v)=0 for all v in V,

and define the right-kernel similarly.

Question: what relation is there between the two kernels , as subspaces
of V?

I am pretty sure the answer has to see with the tensor product V(x)V;

but I am not sure of how to express the dual of VxV in terms of the tensor

product. Any ideas?

Thanks.

2. Apr 22, 2011

### Bacle

Unfortunately, the quote function is not working well, so I will
improvise. Let n be the dimension of V, so that Dim(VxV)=2n
and Dim(V(x)V)=n^2

I know that the element B' in V(x)V , corresponding to the
bilinear map B : VxV-->k , is a linear functional in V(x)V, and so
the kernel of B' has codimension 1, or, equivalently, dimension
n^2-1 in B(x)B.

But I don't know any properties of kernels of bilinear maps, and
I don't know if there is a way of somehow pulling back the kernel
of B' back into the kernel of B in VxV.

Anyway, I'll keep trying. Any Advice Appreciated.