Distinction between meanings of Euclidean space transformations
Geometric transformations can be distinguished into two types: active or alibi transformations which change the physical position of a set of points relative to a fixed frame of reference or coordinate system ("alibi" meaning "being somewhere else at the same time"); and passive or alias transformations which leave points fixed but change the frame of reference or coordinate system relative to which they are described ("alias" meaning "going under a different name"). By "transformation", mathematicians usually refer to active transformations, while physicists and engineers could mean either.
For instance, active transformations are useful to describe successive positions of a rigid body. On the other hand, passive transformations may be useful in human motion analysis to observe the motion of the tibia relative to the femur, that is, its motion relative to a ("local") coordinate system which moves together with the femur, rather than a ("global") coordinate system which is fixed to the floor.
In three-dimensional Euclidean space, any proper rigid transformation, whether active or passive, can be represented as a screw displacement, the composition of a translation along an axis and a rotation about that axis.
The terms "active transformation" and "passive transformation" were first introduced in 1957 by Valentine Bargmann for describing Lorentz transformations in special relativity.
Example.
As an example, let the vector formula_1, be a vector in the plane. A rotation of the vector through an angle "θ" in counterclockwise direction is given by the rotation matrix:
formula_2
which can be viewed either as an "active transformation" or a "passive transformation" (where the above matrix will be inverted), as described below.
Spatial transformations in the Euclidean space R3.
In general a spatial transformation formula_3 may consist of a translation and a linear transformation. In the following, the translation will be omitted, and the linear transformation will be represented by a 3×3 matrix formula_4.
Active transformation.
As an active transformation, formula_4 transforms the initial vector formula_6 into a new vector formula_7.
If one views formula_8 as a new basis, then the coordinates of the new vector formula_9 in the new basis are the same as those of formula_10 in the original basis. Note that active transformations make sense even as a linear transformation into a different vector space. It makes sense to write the new vector in the unprimed basis (as above) only when the transformation is from the space into itself.
Passive transformation.
On the other hand, when one views formula_4 as a passive transformation, the initial vector formula_6 is left unchanged, while the coordinate system and its basis vectors are transformed in the opposite direction, that is, with the inverse transformation formula_13. This gives a new coordinate system "XYZ" with basis vectors:
formula_14
The new coordinates formula_15 of formula_16 with respect to the new coordinate system "XYZ" are given by: 
formula_17
From this equation one sees that the new coordinates are given by
formula_18
As a passive transformation formula_4 transforms the old coordinates into the new ones.
Note the equivalence between the two kinds of transformations: the coordinates of the new point in the active transformation and the new coordinates of the point in the passive transformation are the same, namely
formula_20
In abstract vector spaces.
The distinction between active and passive transformations can be seen mathematically by considering abstract vector spaces. 
Fix a finite-dimensional vector space formula_21 over a field formula_22 (thought of as formula_23 or formula_24), and a basis formula_25 of formula_21. This basis provides an isomorphism formula_27 via the component map formula_28.
An active transformation is then an endomorphism on formula_21, that is, a linear map from formula_21 to itself. Taking such a transformation formula_31, a vector formula_32 transforms as formula_33. The components of formula_34 with respect to the basis formula_35 are defined via the equation formula_36. Then, the components of formula_37 transform as formula_38.
A passive transformation is instead an endomorphism on formula_39. This is applied to the components: formula_40. Provided that formula_4 is invertible, the new basis formula_42 is determined by asking that formula_43, from which the expression formula_44 can be derived.
Although the spaces formula_45 and formula_46 are isomorphic, they are not canonically isomorphic. Nevertheless a choice of basis formula_35 allows construction of an isomorphism.
As left- and right-actions.
Often one restricts to the case where the maps are invertible, so that active transformations are the general linear group formula_48 of transformations while passive transformations are the group formula_49.
The transformations can then be understood as acting on the space of bases for formula_21. An active transformation formula_51 sends the basis formula_52. Meanwhile a passive transformation formula_53 sends the basis formula_54. 
The inverse in the passive transformation ensures the "components" transform identically under formula_34 and formula_4. This then gives a sharp distinction between active and passive transformations: active transformations act from the left on bases, while the passive transformations act from the right, due to the inverse.
This observation is made more natural by viewing bases formula_35 as a choice of isomorphism formula_58. The space of bases is equivalently the space of such isomorphisms, denoted formula_59. Active transformations, identified with formula_48, act on formula_59 from the left by composition, while passive transformations, identified with formula_49 acts on formula_59 from the right by pre-composition.
This turns the space of bases into a "left" formula_48-torsor and a "right" formula_49-torsor.
From a physical perspective, active transformations can be characterized as transformations of physical space, while passive transformations are characterized as redundancies in the description of physical space. This plays an important role in mathematical gauge theory, where gauge transformations are described mathematically by transition maps which act "from the right" on fibers.