 |
| 我的日历 |
| |
|
| 分类日志 |
| |
|
| 友情链接 |
|
|
| 最新评论 |
|
|
| 搜索日志 |
|
|
| 访问计数 |
|
|
| 获取 RSS |
| |
 | |
| |
向量空间张量积[Vector Space Tensor Product] [2005-7-30]
iamet 发表在 ∑〖数学〗
| The tensor product of two vector spaces  and  , denoted  and also called the tensor direct product, is a way of creating a new vector space analogous to multiplication of integers. For instance,
 (1)
In particular,
 (2)
Also, the tensor product obeys a distributive law with the direct sum operation:
 (3)
The analogy with an algebra is the motivation behind K-theory. The tensor product of two tensors  and can be implemented in Mathematica as
TensorProduct[a_List, b_List] := Outer[List, a, b]
Algebraically, the vector space  is spanned by elements of the form  , and the following rules are satisfied, for any scalar  . The definition is the same no matter which scalar field is used.
 (4)
 (5)
 (6)
One basic consequence of these formulas is that
 (7)
A vector basis  of  and  of  gives a basis for  , namely  , for all pairs  . An arbitrary element of  can be written uniquely as  , where  are scalars. If  is  dimensional and  is  dimensional, then  has dimension  .
Using tensor products, one can define symmetric tensors, antisymmetric tensors, as well as the exterior algebra. Moreover, the tensor product is generalized to the vector bundle tensor product. In particular, tensor products of the tangent bundle and its dual bundle are studied in Riemannian geometry and physics. Sections of these bundles are often called tensors. In addition, it is possible to take the representation tensor product to get another representation.
All of these versions of tensor product can be understood as module tensor products. The trick is to find the right way to think of these spaces as modules. |
学好数理化，走遍天下都不怕！ | |
|
|
