Active and passive transformation: Difference between revisions

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[[File:Alias and alibi rotations.png|thumb|upright=1.8|Rotation considered as a passive (''alias'') or active (''alibi'') transformation]]
[[File:Alias and alibi transformations 1 en.png|thumb|upright=1.8|Translation and rotation as passive (''alias'') or active (''alibi'') transformations]]
As an example, let the vector <math>\mathbf{v}=(v_1,v_2)\in \R^2</math>, be a vector in the plane. A rotation of the vector through an angle θ in counterclockwise direction is given by the [[rotation matrix]]:
:<math>R=
\begin{pmatrix}
===Active transformation===
As an active transformation, ''R'' rotates the initial vector '''v''', and a new vector '''v'''' is obtained. For a counterclockwise rotation of '''v''' with respect to the fixed coordinate system:
:<math>\mathbf{v'}'=R\mathbf{v}=\begin{pmatrix}
\cos \theta & -\sin \theta\\
\sin \theta & \cos \theta
\end{pmatrix}.</math>
 
If one views <math>\{\mathbf{e}'_1=R(1,0),\ \mathbf{e}'_2=R(0,1)\}</math> as a new basis, then the coordinates of the new vector <math>\mathbf{v}'=v_1\mathbf{e}'_1+v_2\mathbf{e}'v′'''_2</math> in the new basis are the same as those of '''<math>\mathbf{v'''}=v_1\mathbf{e}_1+v_2\mathbf{e}_2</math> 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 ===
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