Understanding Tensor Components: The Role of Index Order in Vector Spaces

[quasar987]

Say V is a vector space with base {e_i}, V* is it's dual with dual basis {e^i}. If someone says that $T^i_{ \ j}$ are the components of a tensor, then I know this means that the actual tensor is

$$\mathbf{T}=T^i_{ \ j}e_i\otimes e^j$$

The order of the indices of the components of T indicates on which set is T acting. In this case, V* x V. Were the components $T_j^{ \ i}$, T would have acted on V x V*.

Now my question.

If $\Gamma$ is a function from vector spaces V to W of respective bases {$e_i$} and {$\tilde{e}_i$}, and if we define the components of $\Gamma$ as the numbers $\Gamma_i^{ \ j}$ such that

[tex]\Gamma(e_i)=\Gamma_i^{ \ j}\tilde{e}_j[/itex],<br /> <br /> is there a meaning to the order of the indiced, or could I have just as well noted the coefficients as $\Gamma^{j}_{ \ i}$?<br /> <br /> Thanks.[/tex]

 

[Chris Hillman]

Hi, quasar987, have you looked at Geroch, Geometry of Physics or Nakayama, Geometry, Topology and Physics? These should answer your questions.

 

[George Jones]

Hi, quasar987, have you looked at Geroch, Geometry of Physics ...

Frankel?

 

[coalquay404]

Frankel?

Presumably, although Geroch's Mathematical Physics, if I recall correctly, also discusses this.

 

[dextercioby]

Hi, quasar987, have you looked at Geroch, Geometry of Physics or Nakayama, Geometry, Topology and Physics? These should answer your questions.

Nakahara, maybe ?

Daniel.