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Bilinear maps: showing nondegenerate in second variable

  1. Jul 23, 2011 #1
    1. The problem statement, all variables and given/known data

    Let V be a n-dimensional vector space.
    Let B: V* x V -> F (field) be the bilinear map defined by B(alpha,v) = alpha(v). Show that B in nondegenerate in the second variable. (4 marks)

    2. Relevant equations

    I know that B is non-degenerate in the second variable if; B(alpha,v)=0 for every alpha a member of V* which implies that v=0.

    3. The attempt at a solution

    My attempt was writing out the definition I know (applying it to the variables given). I do not know how to show that it is true for the bilinear map given.

    The other thing I know, but not sure if it helps or not, is that if B is symmetric/skew then we can define the radical so that B is non-degenerate on each variable iff radB={0}. But I'm not sure if I need this or not.

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    I am currently working through some of the past papers I did for my second year units at university, trying to understand the things I couldn't do at the time ready for my third year units. However obviously the university hasn't supplied us with answers yet so I cannot check what I am doing is correct or not. Thank you for your help!

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  2. jcsd
  3. Jul 24, 2011 #2

    lavinia

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    Science Advisor
    Gold Member

    I guess I am confused about the question.
    A non zero linear functional must by definition evaluate to a non-zero number for some vector. No?
     
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