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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
\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.
\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
\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.
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