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

by Bacle
Tags: bilinear, bvxv>k, kernels, maps, vspaces
 P: 662 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.
 P: 662 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.

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