Active and passive transformation: Difference between revisions

Revision as of 16:14, 10 October 2022

Revision as of 08:37, 28 September 2023

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An '''active transformation''' is then an [[endomorphism]] on <math>V</math>, that is, a linear map from <math>V</math> to itself. Taking such a transformation <math>\tau \in \text{End}(V)</math>, a vector <math>v \in V</math> transforms as <math>v \mapsto \tau v</math>. The components of <math>\tau</math> with respect to the basis <math>\mathcal{B}</math> are defined via the equation <math display="inline">\tau e_i = \sum_j\tau_{ji}e_j</math>. Then, the components of <math>v</math> transform as <math>v_i \mapsto \tau_{ij}v_j</math>.

A '''passive transformation''' is instead an endomorphism on <math>K^n</math>. This is applied to the components: <math>v_i \mapsto T_{ij}v_j =: v'_i</math>. TheProvided that <math>T</math> is invertible, the new basis <math>\mathcal{B}' = \{e'_i\}</math> is determined by asking that <math>v_ie_i = v'_i e'_i</math>, from which the expression <math>e'_i = (T^{-1})_{ji}e_j</math> can be derived.

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