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In my lecture notes on classical electrodynamics, we are introduced to the Lorentz transformations: a system S' moves relative to a system S with positive veloticy v in the x axis (meassured in S), spacial axis are parallel, origin of times t and t' coincide, etc., which gives the well-known transformation of coordinates of one relative to the other: http://www.relativitycalculator.com...cities/Lorentz_Equations_in_S_and_S-prime.png (sorry, I don't know how to write equations here).

As I understand, this would be a passive transformation. This transforms the same event from S coordinates to S' coordinates.

However, some pages later it says: "

*Active*Lorentz boosts transform the (x, y, z) spacial coordinates and instant t in which it happens into (x', y', z') and t' given in the same reference system S with the expresions:" and proceeds to write the same expressions as before EXCEPT the sign of v is changed. So like if the primed and non-primed were exchanged in the picture. It further states that the change of sign is due to the active nature of these transformations as opposed to the passive one of those.

But I'm not getting why. As I understand it, an active transformation can be interpreted as bringing what another system S'

*sees*to our system S; either way, the components are the same. Obviously, this doesn't match the change in sign in the velocity. Any help?