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While studying dual vectors in general relativity, it was written as we all know that dual vectors (under Lorentz Transformation) transform as follows:

[itex]\tilde{u}[/itex][itex]_{a}[/itex] = [itex]\Lambda[/itex][itex]^{b}_{a}[/itex]μ[itex]_{b}[/itex]

where [itex]\Lambda[/itex][itex]^{b}_{a}[/itex]= η[itex]_{ac}[/itex]L[itex]^{c}[/itex][itex]_{d}[/itex]η[itex]^{db}[/itex]

I was wondering if one can prove the latter or we take it as is.

This to a certain extend can be related to [itex]\Lambda[/itex] = ηLη[itex]^{-1}[/itex], so is it that they took this relation and placed indices in a way if they are summed over we get [itex]\Lambda[/itex][itex]^{b}_{a}[/itex]? Or is there any clearer procedure?

Thanks!

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# Transpose Inverse Property (Dual Vectors)

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