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Bilinear maps and inner product

  1. Dec 19, 2009 #1
    1. The problem statement, all variables and given/known data
    We are given a linear map f, f:R2xR2->R. f has the following properties:
    1)It is linear for the changes of the first variable
    2)It is linear for the changes of the second variable
    3)f((3,8),(3,8))=13

    We are asked to say if f is anyhow related to the normal inner product of R2

    2. Relevant equations



    3. The attempt at a solution

    I know that the inner product of R2 is a bilinear map like f. If f actually WAS the linear product then it should be f((3,8),(3,8))=73. Perhaps I could say that f is the inner product, multiplied by some value? I dont really understand the question, but I have posted it exactly as stated.
     
  2. jcsd
  3. Dec 20, 2009 #2

    HallsofIvy

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    You mean if it really were the inner product. A bilinear map can have any value at all on pairs of basis vectors and then is determined for all other pairs. In particular, a bilinear map on R2 must be of the form f((a,b)(c,d))= k1ac+ k2bc+ k3ad+ k4bd. The "normal inner product of R2" is the specific bilinear form f((a,b),(c,d)= ac+ bd. In other words, k1= k4= 1 and k2= k3= 0.

    Here, we know only that f((3,8),(3,8))= 9k1+ 24(k2+ k3)+ 64k4= 13. There are, of course, an infinite number of choices for the k's that would give that. I see know way, on the basis of a single value, to say that it is "related" to the inner product any more than any other bilinear map,
     
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