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Exercise:
Let VW be vector spaces over K and T:Vlongrightarrow W a linear map. Define a new map overlineT: V/textKerTlongrightarrow textImT as follows. Let xin V/textKerT. Choose a representative vin V for x i.e. xv. Define overlineTx:Tv. Then: abcliste abc overlineT is well defined and is a linear map. abc overlineT: V/textKerT longrightarrow textImT is an isomorphism. abcliste

Solution:
Proof. abcliste abc overlineT is well defined. Suppose xvv' we need to show that TvTv'. Indeed vv' implies that v-v'in textKerT Longrightarrow Tv-v' Longrightarrow TvTv'. abc overlineT is linear. overlineTalpha voverlineTalpha v Talpha v alpha Tv alpha overlineTv overlineTv_+v_overlineTv_+v_Tv_+v_Tv_+Tv_Tv_+Tv_. overlineT is injective. Let xin textKeroverlineT. Write x as xv for some vin V. Then overlineTxTv Longrightarrow vin textKerT Longrightarrow xv_textV/KerT. overlineT is surjective. Let win textImT Longrightarrow exists vin V s.t. Tvw Longrightarrow overlineTvTvw. This shows that textImTsubseteq textImoverlineT Longrightarrow textImoverlineTtextImT. abcliste
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Exercise:
Let VW be vector spaces over K and T:Vlongrightarrow W a linear map. Define a new map overlineT: V/textKerTlongrightarrow textImT as follows. Let xin V/textKerT. Choose a representative vin V for x i.e. xv. Define overlineTx:Tv. Then: abcliste abc overlineT is well defined and is a linear map. abc overlineT: V/textKerT longrightarrow textImT is an isomorphism. abcliste

Solution:
Proof. abcliste abc overlineT is well defined. Suppose xvv' we need to show that TvTv'. Indeed vv' implies that v-v'in textKerT Longrightarrow Tv-v' Longrightarrow TvTv'. abc overlineT is linear. overlineTalpha voverlineTalpha v Talpha v alpha Tv alpha overlineTv overlineTv_+v_overlineTv_+v_Tv_+v_Tv_+Tv_Tv_+Tv_. overlineT is injective. Let xin textKeroverlineT. Write x as xv for some vin V. Then overlineTxTv Longrightarrow vin textKerT Longrightarrow xv_textV/KerT. overlineT is surjective. Let win textImT Longrightarrow exists vin V s.t. Tvw Longrightarrow overlineTvTvw. This shows that textImTsubseteq textImoverlineT Longrightarrow textImoverlineTtextImT. abcliste
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eth, hs22, isomorphism, lineare algebra, proof
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