# Bilinear maps: showing nondegenerate in second variable

1. Jul 23, 2011

### MimuiSnoopy

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. Jul 24, 2011

### lavinia

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?