Geometric transformations can be distinguished into two types:active oralibi transformations which change the physical position of a set ofpoints relative to a fixedframe of reference orcoordinate system (alibi meaning "being somewhere else at the same time"); andpassive oralias 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").^[1][2] Bytransformation,mathematicians usually refer to active transformations, whilephysicists andengineers could mean either.^{[citation needed]}

For instance, active transformations are useful to describe successive positions of arigid body. On the other hand, passive transformations may be useful in human motion analysis to observe the motion of thetibia relative to thefemur, 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.^[2]

Inthree-dimensional Euclidean space, anyproper rigid transformation, whether active or passive, can be represented as ascrew displacement, the composition of atranslation along an axis and arotation about that axis.

The termsactive transformation andpassive transformation were first introduced in 1957 byValentine Bargmann for describingLorentz transformations inspecial relativity.^[3]

As an example, let the vector${\displaystyle \mathbf{v}=(v_1,v_2)\in {\mathbb{R}}^2}$, be a vector in the plane. A rotation of the vector through an angleθ in counterclockwise direction is given by therotation matrix:which can be viewed either as anactive transformation or apassive transformation (where the abovematrix will beinverted), as described below.

In general a spatial transformation${\displaystyle T:{\mathbb{R}}^3\to {\mathbb{R}}^3}$ 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${\displaystyle T}$.

As an active transformation,${\displaystyle T}$ transforms the initial vector${\displaystyle \mathbf{v}=(v_x,v_y,v_z)}$ into a new vector${\displaystyle {\mathbf{v}}^{\prime }=(v_x^{\prime },v_y^{\prime },v_z^{\prime })=T\mathbf{v}=T(v_x,v_y,v_z)}$.

If one views${\displaystyle \{{\mathbf{e}}_x^{\prime }=T(1,0,0),{\mathbf{e}}_y^{\prime }=T(0,1,0),{\mathbf{e}}_z^{\prime }=T(0,0,1)\}}$ as a newbasis, then the coordinates of the new vector${\displaystyle {\mathbf{v}}^{\prime }=v_x{\mathbf{e}}_x^{\prime }+v_y{\mathbf{e}}_y^{\prime }+v_z{\mathbf{e}}_z^{\prime }}$ in the new basis are the same as those of${\displaystyle \mathbf{v}=v_x{\mathbf{e}}_x+v_y{\mathbf{e}}_y+v_z{\mathbf{e}}_z}$ in the original basis. Note that active transformations make sense even as a linear transformation into a differentvector 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.

On the other hand, when one views${\displaystyle T}$ as a passive transformation, the initial vector${\displaystyle \mathbf{v}=(v_x,v_y,v_z)}$ is left unchanged, while the coordinate system and its basis vectors are transformed in the opposite direction, that is, with the inverse transformation${\displaystyle T^{-1}}$.^[4] This gives a new coordinate systemXYZ with basis vectors:$${\displaystyle {\mathbf{e}}_X=T^{-1}(1,0,0),{\mathbf{e}}_Y=T^{-1}(0,1,0),{\mathbf{e}}_Z=T^{-1}(0,0,1)}$$

The new coordinates${\displaystyle (v_X,v_Y,v_Z)}$ of${\displaystyle \mathbf{v}}$ with respect to the new coordinate systemXYZ are given by:$${\displaystyle \mathbf{v}=(v_x,v_y,v_z)=v_X{\mathbf{e}}_X+v_Y{\mathbf{e}}_Y+v_Z{\mathbf{e}}_Z=T^{-1}(v_X,v_Y,v_Z).}$$

From this equation one sees that the new coordinates are given by$${\displaystyle (v_X,v_Y,v_Z)=T(v_x,v_y,v_z).}$$

As a passive transformation${\displaystyle T}$ 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$${\displaystyle (v_X,v_Y,v_Z)=(v_x^{\prime },v_y^{\prime },v_z^{\prime }).}$$

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The distinction between active and passive transformations can be seen mathematically by considering abstractvector spaces.

Fix a finite-dimensional vector space${\displaystyle V}$ over a field${\displaystyle K}$ (thought of as${\displaystyle \mathbb{R}}$ or${\displaystyle \mathbb{C}}$), and a basis${\displaystyle \mathcal{B}=\{e_i{\}}_{1\le i\le n}}$ of${\displaystyle V}$. This basis provides an isomorphism${\displaystyle C:K^n\to V}$ via the component map$(v_i{)}_{1\le i\le n}=(v_1,\cdots ,v_n)\mapsto \sum _i{v}_i{e}_i$.

Anactive transformation is then anendomorphism on${\displaystyle V}$, that is, a linear map from${\displaystyle V}$ to itself. Taking such a transformation${\displaystyle \tau \in \text{End}(V)}$, a vector${\displaystyle v\in V}$ transforms as${\displaystyle v\mapsto \tau v}$. The components of${\displaystyle \tau }$ with respect to the basis${\displaystyle \mathcal{B}}$ are defined via the equation$\tau e_i=\sum _j{\tau }_{ji}{e}_j$. Then, the components of${\displaystyle v}$ transform as${\displaystyle v_i\mapsto {\tau }_{ij}{v}_j}$.

Apassive transformation is instead an endomorphism on${\displaystyle K^n}$. This is applied to the components:${\displaystyle v_i\mapsto T_{ij}{v}_j=:v_i^{\prime }}$. Provided that${\displaystyle T}$ is invertible, the new basis${\displaystyle {\mathcal{B}}^{\prime }=\{e_i^{\prime }\}}$ is determined by asking that${\displaystyle v_i{e}_i=v_i^{\prime }{e}_i^{\prime }}$, from which the expression${\displaystyle e_i^{\prime }=(T^{-1}{)}_{ji}{e}_j}$ can be derived.

Although the spaces${\displaystyle \text{End}(V)}$ and${\displaystyle \text{End}(K^n)}$ are isomorphic, they are not canonically isomorphic. Nevertheless a choice of basis${\displaystyle \mathcal{B}}$ allows construction of an isomorphism.

Often one restricts to the case where the maps are invertible, so that active transformations are thegeneral linear group${\displaystyle \text{GL}(V)}$ of transformations while passive transformations are the group${\displaystyle \text{GL}(n,K)}$.

The transformations can then be understood as acting on the space of bases for${\displaystyle V}$. An active transformation${\displaystyle \tau \in \text{GL}(V)}$ sends the basis${\displaystyle \{e_i\}\mapsto \{\tau e_i\}}$. Meanwhile a passive transformation${\displaystyle T\in \text{GL}(n,K)}$ sends the basis$\{e_i\}\mapsto \left\{\sum _j(T^{-1}{)}_{ji}{e}_j\right\}$.

The inverse in the passive transformation ensures thecomponents transform identically under${\displaystyle \tau }$ and${\displaystyle T}$. This then gives a sharp distinction between active and passive transformations: active transformationsact 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${\displaystyle \mathcal{B}}$ as a choice of isomorphism${\displaystyle {\mathrm{\Phi }}_{\mathcal{B}}:K^n\to V}$. The space of bases is equivalently the space of such isomorphisms, denoted${\displaystyle \text{Iso}(K^n,V)}$. Active transformations, identified with${\displaystyle \text{GL}(V)}$, act on${\displaystyle \text{Iso}(K^n,V)}$ from the left by composition, that is if${\displaystyle \tau }$ represents an active transformation, we have${\displaystyle {\mathrm{\Phi }}_{{\mathcal{B}}^{\prime }}=\tau \circ {\mathrm{\Phi }}_{\mathcal{B}}}$. On the opposite, passive transformations, identified with${\displaystyle \text{GL}(n,K)}$ acts on${\displaystyle \text{Iso}(K^n,V)}$ from the right by pre-composition, that is if${\displaystyle T}$ represents a passive transformation, we have${\displaystyle {\mathrm{\Phi }}_{{\mathcal{B}}^{″}}={\mathrm{\Phi }}_{\mathcal{B}}\circ T}$.

This turns the space of bases into aleft${\displaystyle \text{GL}(V)}$-torsor and aright${\displaystyle \text{GL}(n,K)}$-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 mathematicalgauge theory, wheregauge transformations are described mathematically by transition maps which actfrom the right on fibers.

• Change of basis
• Covariance and contravariance of vectors
• Rotation of axes
• Translation of axes

• Dirk Struik (1953)Lectures on Analytic and Projective Geometry, page 84,Addison-Wesley.

• UI ambiguity

• Systems theory
• Mathematical terminology
• Concepts in physics

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