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

  1. Apr 21, 2011 #1
    Hi, all:

    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
    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?

  2. jcsd
  3. Apr 22, 2011 #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
    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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