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

  1. 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. jcsd
  3. 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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