Active and passive transformation

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For the concept of "passive transformation" in grammar, see active voice and passive voice.

In the physical sciences, an active transformation is one which actually changes the physical state of a system, and makes sense even in the absence of a coordinate system whereas a passive transformation is merely a change in the coordinate system of no physical significance. The distinction between active and passive transformations is one which should always be kept in mind when working with transformations. By default, by transformation, mathematicians usually mean passive transformations, while physicists could mean either.

As an example, in the vector space R², let {e₁,e₂} be a basis, and consider the vector v = v¹e₁ + v²e₂. Rotation of the plane is given by the matrix

$R= \begin{pmatrix} \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{pmatrix},$

which can be viewed either as an active transformation or a passive transformation.

As an active transformation, R rotates all vectors, including v and the basis vectors e₁ and e₂. Thus a new vector v' is obtained:

$\mathbf{v}'=R\mathbf{v}=v^1R\mathbf{e}_1+v^2R\mathbf{e}_2.$

If one views {Re₁,Re₂} as a new basis, then the components of the new vector v' remain the same in the new (primed) basis, because linear transformations do not act on scalars, and the components of a vector are simply scalars. One may also calculate the components of the new vector in terms of the old (unprimed) basis, in which case the formula

$(R\mathbf{v})^b=R_a{}^bv^a$

is obtained. But note that active transformations make sense even when it is a linear transformation into a different vector space. It only makes sense to write the primed vector in the unprimed basis when the transformation is from the space into itself.

On the other hand, when one views R as a passive transformation, the vector v is left unchanged, while the basis vectors get rotated. In order for the vector to remain fixed, the components in terms of the new basis must also change.

$\mathbf{v}=v^a\mathbf{e}_a=v'^aR\mathbf{e}_a$

From this equation one sees that the new components with respect to the new coordinates are given by

v' a = (R - 1) b a v b

so that

$\mathbf{v}=v'^a\mathbf{e}'_a=v^b(R^{-1})_b{}^aR_a{}^c\mathbf{e}_c=v^b\mathbf{e}_b.$

Thus, in order for the vector to remain unchanged by the passive transformation, the components of the vector have to transform, and according to the inverse of the transformation operator.