- #1
Bacle
- 662
- 1
Hi, Everyone:
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.
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.