Active and passive transformation: Difference between revisions

Revision as of 09:00, 8 November 2024

Latest revision as of 16:18, 24 February 2025

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The inverse in the passive transformation ensures the ''components'' transform identically under <math>\tau</math> and <math>T</math>. This then gives a sharp distinction between active and passive transformations: active transformations [[left action|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 <math>\mathcal{B}</math> as a choice of isomorphism <math>\Phi_{\mathcal{B}}: VK^n \rightarrow K^nV</math>. The space of bases is equivalently the space of such isomorphisms, denoted <math>\text{Iso}(V, K^n, V)</math>. Active transformations, identified with <math>\text{GL}(V)</math>, act on <math>\text{Iso}(V, K^n, V)</math> from the left by composition, whilethat is if <math>\tau</math> represents an active transformation, we have <math>\Phi_{\mathcal{B'}} = \tau \circ \Phi_{\mathcal{B}}</math>. On the opposite, passive transformations, identified with <math>\text{GL}(n, K)</math> acts on <math>\text{Iso}(V, K^n, V)</math> from the right by pre-composition, that is if <math>T</math> represents a passive transformation, we have <math>\Phi_{\mathcal{B''}} = \Phi_{\mathcal{B}} \circ T</math>.

This turns the space of bases into a ''left'' <math>\text{GL}(V)</math>-[[torsor]] and a ''right'' <math>\text{GL}(n, K)</math>-torsor.