I am trying to understand what each transformation means and what their differences are but many books that don't state which transformation they are referring to make it a bit confusing to understand which is which.
Aside from what their differences are, I want to particularly know how do matrices change when we deal with each transformation, so if anybody could help it would be really appreciated.
Also, which one is used by physicists the most and why?

EDIT: An example would be a great way to understand these concepts

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  • $\begingroup$ Crossposted to math.stackexchange.com/q/1844714/11127 Related: physics.stackexchange.com/q/116856/2451 and links therein. $\endgroup$ – Qmechanic Jun 30 '16 at 13:21
  • $\begingroup$ Active transformation: you transform a state (acting on the physical system). Passive transformation: you change the coordinate that you use to describe the system. $\endgroup$ – ophelia Jun 30 '16 at 19:45
  • $\begingroup$ @ophelia Could you please write it as an answer along with the mathematics? $\endgroup$ – TheQuantumMan Jun 30 '16 at 19:46
  • $\begingroup$ Which one is popular: (personal feelings), in the older time passive transformation is more often used; in the more recent time, active transformation seems to be used often (say, in Sakurai's text). And I guess the postmodern text would again use passive transformation :-) $\endgroup$ – ophelia Jun 30 '16 at 19:50
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    $\begingroup$ @Qmechanic the approach of mathematicians is a bit different(more rigorous) than that of most physicsts. $\endgroup$ – TheQuantumMan Jul 1 '16 at 14:22

Here is what I understand: if you have a particle at state $|x \rangle$, active translating it by $a$ means moving the particle to state $ | x + a \rangle$. Passive transformation means you keep the particle in the same place, and change the coordinate by new variable $x = x' + a$ (note that the coordinate system is translated backwards $-a$).

I am not very sure which one is popular. As I commented, here are my personal feelings: in the older time passive transformation is often used; in the more recent time, active transformation seems to be used more often (say, in Sakurai's text). And I guess in the next generation, textbooks would again use passive transformation.

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    $\begingroup$ So the active transformation means physically moving a free particle with time and observed from the same still reference frame, for example, if $a=vt$. And the passive transformation means physically observing the same particle at the same time $t$ from two different reference frames. $\endgroup$ – Vladimir Kalitvianski Jun 30 '16 at 20:20
  • $\begingroup$ No, there is no such notion of time mentioned here. Say like this: to describe the position of a cat, you say it is 1 m far from you. Here you are the origin of the coordinate system, and the cat is the "particle". Active transforming the cat means bringing the cat, say 0.5 m forward, so that it is 1.5 m from you. Passive transformation means that you go backward 0.5 m and now the cat is also 1.5 m away from you. @VladimirKalitvianski $\endgroup$ – ophelia Jun 30 '16 at 20:28
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    $\begingroup$ You are wrong. Any forced movement is subject to the corresponding equations since the result strongly depends on the force. $\endgroup$ – Vladimir Kalitvianski Jun 30 '16 at 20:34
  • $\begingroup$ so at active transformations the cat moves while at passive transformations we move(the origin) in the opposite direction,right? $\endgroup$ – TheQuantumMan Jul 1 '16 at 8:33
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    $\begingroup$ @ophelia: We rotate nothing. We may have differently placed observers and our "rotation" formulas recalculate the observation results of one observer to those of another. $\endgroup$ – Vladimir Kalitvianski Jul 6 '16 at 12:45

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