Editing Principal bundle (section)

===Associated vector bundles and frames===
{{See also| Frame bundle}}
If <math>P</math> is a principal <math>G</math>-bundle and <math>V</math> is a [[linear representation]] of <math>G</math>, then one can construct a vector bundle <math>E=P\times_G V</math> with fibre <math>V</math>, as the quotient of the product <math>P</math>&times;<math>V</math> by the diagonal action of <math>G</math>. This is a special case of the [[associated bundle]] construction, and <math>E</math> is called an [[associated vector bundle]] to <math>P</math>. If the representation of <math>G</math> on <math>V</math> is [[faithful representation|faithful]], so that <math>G</math> is a subgroup of the general linear group GL(<math>V</math>), then <math>E</math> is a <math>G</math>-bundle and <math>P</math> provides a reduction of structure group of the frame bundle of <math>E</math> from <math>GL(V)</math> to <math>G</math>. This is the sense in which principal bundles provide an abstract formulation of the theory of frame bundles.