- #1

AndersF

- 27

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- TL;DR Summary
- Why can we express the dual basis vectors in terms of the original basis vectors through the dual metric tensor in this way? ##\mathbf{e}^i=g^{ij}\mathbf{e}_j##

I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and

##\mathbf{e}^i=g^{ij}\mathbf{e}_j##

##\mathbf{e}_i=g_{ij}\mathbf{e}^j##

What would be the mathematical justification for these expressions?

*vice versa*, in these ways:##\mathbf{e}^i=g^{ij}\mathbf{e}_j##

##\mathbf{e}_i=g_{ij}\mathbf{e}^j##

What would be the mathematical justification for these expressions?