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Reference.org
Faithful representation
open-in-new
<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, especially in an area of <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a> known as <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a>, a faithful representation ρ of a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> G on a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> V is a <a href="/facts/Linear_representation/3ydLtaGH">linear representation</a> in which different elements g of G are represented by distinct <a href="/facts/Linear_mapping/Q1PBKNVV">linear mappings</a> <i>ρ</i>(<i>g</i>). In more abstract language, this means that the <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphism</a> ρ : G → G L ( V ) {\displaystyle \rho :G\to GL(V)} is <a href="/facts/Injective/5ES6tqf5">injective</a> (or <a href="/facts/Injective/5ES6tqf5">one-to-one</a>). </p>