# Rank of a bilinear form

I have to prove that a bilinear form $B$ has full rank, and I would like to know some ideas on how to prove that.

Can anyone give an idea?

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Hint. Show that a bilinear map is non-degenerate if and only if its matrix representation has full rank.

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The problem is that is very hard to find a matrix representation (I am working with a particular bilineal form, but i want some alternative ideas instead of asking for the solution, because i would like to think the problem by my self –  Dimitri Feb 24 '13 at 22:08
@Dimitri Can you prove that it's non-degenerate? (That is $\langle x,y\rangle = 0$ for all $y$ implies that $x=0$.) –  Alexander Gruber Feb 26 '13 at 1:16
Yes, that is what i try to do, with a little help and work i think i almost have it :) –  Dimitri Feb 26 '13 at 1:20

So what I've long time known as Gaussian reduction (not Gaussian elimination, other technique) is now Lagrange's reduction... Maybe it's me. I'm sure there is another name more popular in the english speaking world. And I'm sure someone will tell us.

Anyway, here is a place where you can find an example, if not a full theory. Note that it is based on the square completion method.

And here is another one, with just enough theory.

Once you have performed the reduction. Count the nonzero coefficients. It is your rank.

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