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Active and passive transformation
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<h2 id="example">Example</h2> <p>As an example, let the vector v = ( v 1 , v 2 ) ∈ R 2 {\displaystyle \mathbf {v} =(v_{1},v_{2})\in \mathbb {R} ^{2}} , be a vector in the plane. A rotation of the vector through an angle <i>θ</i> in counterclockwise direction is given by the <a href="/facts/Rotation_matrix/c2K2Ftl8">rotation matrix</a>: R = ( cos θ − sin θ sin θ cos θ ) , {\displaystyle R={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}},} which can be viewed either as an <i>active transformation</i> or a <i>passive transformation</i> (where the above <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> will be <a href="/facts/Inverse_matrix/fPqXk3V8">inverted</a>), as described below. </p> <h2 id="spatial-transformations-in-the-euclidean-space-r3">Spatial transformations in the Euclidean space R3</h2> <p>In general a spatial transformation T : R 3 → R 3 {\displaystyle T\colon \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 T {\displaystyle T} . </p> <h3>Active transformation</h3> <p>As an active transformation, T {\displaystyle T} transforms the initial vector v = ( v x , v y , v z ) {\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z})} into a new vector v ′ = ( v x ′ , v y ′ , v z ′ ) = T v = T ( v x , v y , v z ) {\displaystyle \mathbf {v} '=(v'_{x},v'_{y},v'_{z})=T\mathbf {v} =T(v_{x},v_{y},v_{z})} . </p><p>If one views { e x ′ = T ( 1 , 0 , 0 ) , e y ′ = T ( 0 , 1 , 0 ) , e z ′ = T ( 0 , 0 , 1 ) } {\displaystyle \{\mathbf {e} '_{x}=T(1,0,0),\ \mathbf {e} '_{y}=T(0,1,0),\ \mathbf {e} '_{z}=T(0,0,1)\}} as a new <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a>, then the coordinates of the new vector v ′ = v x e x ′ + v y e y ′ + v z e z ′ {\displaystyle \mathbf {v} '=v_{x}\mathbf {e} '_{x}+v_{y}\mathbf {e} '_{y}+v_{z}\mathbf {e} '_{z}} in the new basis are the same as those of v = v x e x + v y e y + v z e z {\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 different <a href="/facts/Vector_space/M1pxMLx2">vector space</a>. It makes sense to write the new vector in the unprimed basis (as above) only when the transformation is from the space into itself. </p> <h3>Passive transformation</h3> <p>On the other hand, when one views T {\displaystyle T} as a passive transformation, the initial vector v = ( v x , v y , v z ) {\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 T − 1 {\displaystyle T^{-1}} .<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> This gives a new coordinate system <i>XYZ</i> with basis vectors: e X = T − 1 ( 1 , 0 , 0 ) , e Y = T − 1 ( 0 , 1 , 0 ) , e Z = T − 1 ( 0 , 0 , 1 ) {\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)} </p><p>The new coordinates ( v X , v Y , v Z ) {\displaystyle (v_{X},v_{Y},v_{Z})} of v {\displaystyle \mathbf {v} } with respect to the new coordinate system <i>XYZ</i> are given by: v = ( v x , v y , v z ) = v X e X + v Y e Y + v Z e Z = T − 1 ( v X , v Y , v Z ) . {\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}).} </p><p>From this equation one sees that the new coordinates are given by ( v X , v Y , v Z ) = T ( v x , v y , v z ) . {\displaystyle (v_{X},v_{Y},v_{Z})=T(v_{x},v_{y},v_{z}).} </p><p>As a passive transformation T {\displaystyle T} transforms the old coordinates into the new ones. </p><p>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 ( v X , v Y , v Z ) = ( v x ′ , v y ′ , v z ′ ) . {\displaystyle (v_{X},v_{Y},v_{Z})=(v'_{x},v'_{y},v'_{z}).} </p> <h2 id="in-abstract-vector-spaces">In abstract vector spaces</h2> <p>The distinction between active and passive transformations can be seen mathematically by considering abstract <a href="/facts/Vector_spaces/M1pxMLx2">vector spaces</a>. </p><p>Fix a finite-dimensional vector space V {\displaystyle V} over a field K {\displaystyle K} (thought of as R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } ), and a basis B = { e i } 1 ≤ i ≤ n {\displaystyle {\mathcal {B}}=\{e_{i}\}_{1\leq i\leq n}} of V {\displaystyle V} . This basis provides an isomorphism C : K n → V {\displaystyle C:K^{n}\rightarrow V} via the component map ( v i ) 1 ≤ i ≤ n = ( v 1 , ⋯ , v n ) ↦ ∑ i v i e i {\textstyle (v_{i})_{1\leq i\leq n}=(v_{1},\cdots ,v_{n})\mapsto \sum _{i}v_{i}e_{i}} . </p><p>An active transformation is then an <a href="/facts/Endomorphism/s7NNlSea">endomorphism</a> on V {\displaystyle V} , that is, a linear map from V {\displaystyle V} to itself. Taking such a transformation τ ∈ End ( V ) {\displaystyle \tau \in {\text{End}}(V)} , a vector v ∈ V {\displaystyle v\in V} transforms as v ↦ τ v {\displaystyle v\mapsto \tau v} . The components of τ {\displaystyle \tau } with respect to the basis B {\displaystyle {\mathcal {B}}} are defined via the equation τ e i = ∑ j τ j i e j {\textstyle \tau e_{i}=\sum _{j}\tau _{ji}e_{j}} . Then, the components of v {\displaystyle v} transform as v i ↦ τ i j v j {\displaystyle v_{i}\mapsto \tau _{ij}v_{j}} . </p><p>A passive transformation is instead an endomorphism on K n {\displaystyle K^{n}} . This is applied to the components: v i ↦ T i j v j =: v i ′ {\displaystyle v_{i}\mapsto T_{ij}v_{j}=:v'_{i}} . Provided that T {\displaystyle T} is invertible, the new basis B ′ = { e i ′ } {\displaystyle {\mathcal {B}}'=\{e'_{i}\}} is determined by asking that v i e i = v i ′ e i ′ {\displaystyle v_{i}e_{i}=v'_{i}e'_{i}} , from which the expression e i ′ = ( T − 1 ) j i e j {\displaystyle e'_{i}=(T^{-1})_{ji}e_{j}} can be derived. </p><p>Although the spaces End ( V ) {\displaystyle {\text{End}}(V)} and End ( K n ) {\displaystyle {\text{End}}({K^{n}})} are isomorphic, they are not canonically isomorphic. Nevertheless a choice of basis B {\displaystyle {\mathcal {B}}} allows construction of an isomorphism. </p> <h3>As left- and right-actions</h3> <p>Often one restricts to the case where the maps are invertible, so that active transformations are the <a href="/facts/General_linear_group/B54gNILS">general linear group</a> GL ( V ) {\displaystyle {\text{GL}}(V)} of transformations while passive transformations are the group GL ( n , K ) {\displaystyle {\text{GL}}(n,K)} . </p><p>The transformations can then be understood as acting on the space of bases for V {\displaystyle V} . An active transformation τ ∈ GL ( V ) {\displaystyle \tau \in {\text{GL}}(V)} sends the basis { e i } ↦ { τ e i } {\displaystyle \{e_{i}\}\mapsto \{\tau e_{i}\}} . Meanwhile a passive transformation T ∈ GL ( n , K ) {\displaystyle T\in {\text{GL}}(n,K)} sends the basis { e i } ↦ { ∑ j ( T − 1 ) j i e j } {\textstyle \{e_{i}\}\mapsto \left\{\sum _{j}(T^{-1})_{ji}e_{j}\right\}} . </p><p>The inverse in the passive transformation ensures the <i>components</i> transform identically under τ {\displaystyle \tau } and T {\displaystyle T} . This then gives a sharp distinction between active and passive transformations: active transformations <a href="/facts/Left_action/Vh9VtUUw">act from the left</a> on bases, while the passive transformations act from the right, due to the inverse. </p><p>This observation is made more natural by viewing bases B {\displaystyle {\mathcal {B}}} as a choice of isomorphism Φ B : K n → V {\displaystyle \Phi _{\mathcal {B}}:K^{n}\rightarrow V} . The space of bases is equivalently the space of such isomorphisms, denoted Iso ( K n , V ) {\displaystyle {\text{Iso}}(K^{n},V)} . Active transformations, identified with GL ( V ) {\displaystyle {\text{GL}}(V)} , act on Iso ( K n , V ) {\displaystyle {\text{Iso}}(K^{n},V)} from the left by composition, that is if τ {\displaystyle \tau } represents an active transformation, we have Φ B ′ = τ ∘ Φ B {\displaystyle \Phi _{\mathcal {B'}}=\tau \circ \Phi _{\mathcal {B}}} . On the opposite, passive transformations, identified with GL ( n , K ) {\displaystyle {\text{GL}}(n,K)} acts on Iso ( K n , V ) {\displaystyle {\text{Iso}}(K^{n},V)} from the right by pre-composition, that is if T {\displaystyle T} represents a passive transformation, we have Φ B ″ = Φ B ∘ T {\displaystyle \Phi _{\mathcal {B''}}=\Phi _{\mathcal {B}}\circ T} . </p><p>This turns the space of bases into a <i>left</i> GL ( V ) {\displaystyle {\text{GL}}(V)} -<a href="/facts/Torsor/N27maCdL">torsor</a> and a <i>right</i> GL ( n , K ) {\displaystyle {\text{GL}}(n,K)} -torsor. </p><p>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 <a href="/facts/Gauge_theory/fSW3gggv">gauge theory</a>, where <a href="/facts/Gauge_transformations/fSW3gggv">gauge transformations</a> are described mathematically by transition maps which act <i>from the right</i> on fibers. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Change_of_basis/Hnu1Spcc">Change of basis</a></li> <li><a href="/facts/Covariance_and_contravariance_of_vectors/s3HGMkhz">Covariance and contravariance of vectors</a></li> <li><a href="/facts/Rotation_of_axes/lgzWdxpn">Rotation of axes</a></li> <li><a href="/facts/Translation_of_axes/3dbW7KB0">Translation of axes</a></li></ul> <ul><li><a href="/facts/Dirk_Struik/D7cVVb9r">Dirk Struik</a> (1953) <i>Lectures on Analytic and Projective Geometry</i>, page 84, <a href="/facts/Addison-Wesley/A7UFATEG">Addison-Wesley</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://web.archive.org/web/20110511051125/https://profiles.google.com/114134834346472219368/posts/AWqcUGXVjcs/Consider-an-equilateral-triangle-in-a-plane-whose">UI ambiguity</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Crampin, M.; Pirani, F.A.E. (1986). Applicable Differential Geometry. Cambridge University Press. p. 22. ISBN 978-0-521-23190-9. <a href="978-0-521-23190-9" target="_blank">978-0-521-23190-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Joseph K. Davidson, Kenneth Henderson Hunt (2004). "§4.4.1 The active interpretation and the active transformation". Robots and screw theory: applications of kinematics and statics to robotics. Oxford University Press. p. 74 ff. ISBN 0-19-856245-4. <a href="0-19-856245-4" target="_blank">0-19-856245-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Joseph K. Davidson, Kenneth Henderson Hunt (2004). "§4.4.1 The active interpretation and the active transformation". Robots and screw theory: applications of kinematics and statics to robotics. Oxford University Press. p. 74 ff. ISBN 0-19-856245-4. <a href="0-19-856245-4" target="_blank">0-19-856245-4</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Bargmann, Valentine (1957). "Relativity". Reviews of Modern Physics. 29 (2): 161–174. Bibcode:1957RvMP...29..161B. doi:10.1103/RevModPhys.29.161. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Amidror, Isaac (2007). "Appendix D: Remark D.12". The theory of the Moiré phenomenon: Aperiodic layers. Springer. p. 346. ISBN 978-1-4020-5457-0. <a href="978-1-4020-5457-0" target="_blank">978-1-4020-5457-0</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>