quasar987

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## Main Question or Discussion Point

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

[tex]\mathbf{T}=T^i_{ \ j}e_i\otimes e^j[/tex]

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 [itex]T_j^{ \ i}[/itex], T would have acted on V x V*.

Now my question.

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

[tex]\Gamma(e_i)=\Gamma_i^{ \ j}\tilde{e}_j[/itex],

is there a meaning to the order of the indiced, or could I have just as well noted the coefficients as [itex]\Gamma^{j}_{ \ i}[/itex]???

Thanks.

[tex]\mathbf{T}=T^i_{ \ j}e_i\otimes e^j[/tex]

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 [itex]T_j^{ \ i}[/itex], T would have acted on V x V*.

Now my question.

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

[tex]\Gamma(e_i)=\Gamma_i^{ \ j}\tilde{e}_j[/itex],

is there a meaning to the order of the indiced, or could I have just as well noted the coefficients as [itex]\Gamma^{j}_{ \ i}[/itex]???

Thanks.

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