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https://texercises.com/exercise/universal-property-of-quotient-spaces/
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Exercise:
Let V be a vector space over K and Usubseteq V a subspace. The quotient space V/U has the following universal property: forall space W and every linear map T:Vlongrightarrow W for which TU i.e. for which U subseteq textKerT exists ! a linear map T':V/U longrightarrow W. Moreover textKerT'textKerT/U.

Solution:
Proof. Similar to the proof of the isomorphism theorem. Define T':V/U longrightarrow W as follows. Let xin V/U. Choose vin V s.t. xv. Define T'x:Tv. Excercise: abcliste abc Show T' is well defined i.e. if vv'Longrightarrow TvTv' abc Show T' is linear. abcliste The diagonal commutes forall vin V T'circ pivT'vTv. Uniqueness of T': Suppose TT'circ pi T''circ pi. We'll show T''T'. Let xin V/U. Since pi is surjective exists vin V s.t. xpiv Longrightarrow TvT'piv and TvT''piv Longrightarrow T'xT''x.
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Exercise:
Let V be a vector space over K and Usubseteq V a subspace. The quotient space V/U has the following universal property: forall space W and every linear map T:Vlongrightarrow W for which TU i.e. for which U subseteq textKerT exists ! a linear map T':V/U longrightarrow W. Moreover textKerT'textKerT/U.

Solution:
Proof. Similar to the proof of the isomorphism theorem. Define T':V/U longrightarrow W as follows. Let xin V/U. Choose vin V s.t. xv. Define T'x:Tv. Excercise: abcliste abc Show T' is well defined i.e. if vv'Longrightarrow TvTv' abc Show T' is linear. abcliste The diagonal commutes forall vin V T'circ pivT'vTv. Uniqueness of T': Suppose TT'circ pi T''circ pi. We'll show T''T'. Let xin V/U. Since pi is surjective exists vin V s.t. xpiv Longrightarrow TvT'piv and TvT''piv Longrightarrow T'xT''x.
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