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?
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?
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 .
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: