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We know that for a fin.dim V.Space , given a basis {v1,..,vn}, then a linear map

T is uniquely defined/specified once we know the values t(v1),T(v2),..,T(vn).

Now, let's consider a bilinear map on VxW (with W not nec. different from V),

both fin. dim. V.spaces over the same field F, with respective bases

{v1,..,vn} and {w1,..,wm}.

I am trying to see what info re the basis vectors of V,W to uniquely determine

a bilinear map defined on VxW. ( we turn VxW into a V.Space over F in the standard

way: basis is {(v1,0),..,(vn,0), (0,w1),..,(0,wm)} , addition is done pairwise, etc.

I know that defining a bilinear map B on the basis alone is not enough to determine

B. What else do we need? I think this has to see with the tensor product V(x)W.

Thanks.