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tensor product basis
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(Theorem)
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The following theorem describes a basis of the tensor product of two vector spaces, in terms of given bases of the component spaces. In passing, it also gives a construction of this tensor product. The exact same method can be used also for free modules over a commutative ring with unit.
Theorem. Let and be vector spaces over a field
with bases
 and 
respectively. Then
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(1) |
is a basis for the tensor product space
.
Proof. Let
this set is obviously a
 -vector-space under pointwise addition and multiplication by scalar (see also this article). Let
 be the bilinear map which satisfies
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(2) |
for all  and  , i.e.,
 is the characteristic function of
 . The reasons ( 2) uniquely defines  on the whole of
 are that
 is a basis of  ,
 is a basis of  , and  is bilinear.
Observe that
is a basis of  . Since one may always define a linear map by giving its values on the basis elements, this implies that there for every
 -vector-space  and every map
 exists a unique linear map
 such that
For  that are bilinear it holds for arbitrary
 and
 that
 , since
As this is the defining property of the tensor product
 however, it follows that  is (an incarnation of) this tensor product, with
 . Hence the claim in the theorem is equivalent to the observation about the basis of  . 
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"tensor product basis" is owned by lars_h.
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Cross-references: equivalent, defining property, map, implies, linear map, bilinear, characteristic function, bilinear map, scalar, multiplication, pointwise addition, field, unit, commutative ring, free modules, vector spaces, basis
There is 1 reference to this entry.
This is version 8 of tensor product basis, born on 2005-07-25, modified 2007-09-01.
Object id is 7254, canonical name is TensorProductBasis.
Accessed 3017 times total.
Classification:
| AMS MSC: | 15A69 (Linear and multilinear algebra; matrix theory :: Multilinear algebra, tensor products) |
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Pending Errata and Addenda
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