|Apr21-11, 09:50 PM||#1|
Left- and Right- Kernels of Bilinear maps B:VxV->K for V.Spaces
Let V be a vector space over k, and let B be a bilinear map into 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
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?
|Apr22-11, 12:27 AM||#2|
Unfortunately, the quote function is not working well, so I will
improvise. Let n be the dimension of V, so that Dim(VxV)=2n
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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