I was told that any bilinear map (where is an n-dimensional vector space and is a field of scalars) can be expressed as , where is an matrix.

Does this statement require proof, and if so, how can I prove it?

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- September 21st 2009, 05:05 PMredsoxfan325Bilinear Maps
I was told that any bilinear map (where is an n-dimensional vector space and is a field of scalars) can be expressed as , where is an matrix.

Does this statement require proof, and if so, how can I prove it? - September 21st 2009, 07:07 PMaliceinwonderland
A bilinear form on V is a bilinear mapping such that

B(u+u', v) = B(u,v) + B(u',v),

B(u, v+v') = B(u,v) + B(u,v'),

B(tu, v) = B(u, tv) = tB(u,v), where t is a scalar in the field F.

Let u= BX, v=BY, where B = (v_1, v_2, ... , v_n) is a basis of V and X,Y are coordinate vectors.

Then, .

Using bilinearity, , where A is . - September 21st 2009, 07:32 PMNonCommAlg
that is actually from to also has no meaning unless you fix a basis for and then by we'll mean the coordinate vectors of with respect to

the basis anyway, the proof is quite easy: define the matrix by now is an vector with in its -th row and is in other rows. it's easy to see

that finally if then using bilinearity of we get: