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https://texercises.com/exercise/kernel-image-injective-surjective/
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Exercise:
Let T:Vlongrightarrow W be a linear map. Then: abcliste abc T is injective iff textKerT_V. abc T is surjective iff textImTW. abcliste

Solution:
Proof. abcliste abc Longrightarrow: Clearly _Vin textKerT. So _Vsubseteq T^-_W. Since T is injective T^-_W contains at most one element Longrightarrow textKerTT^-_W_V. Longleftarrow: Let v_v_in V and ase Tv_Tv_. We need to show that v_v_. Indeed we have Tv_-v_Tv_-Tv_. Longrightarrow v_-v_in textKerT. But by asption textKerT_V. Longrightarrow v_-v_ Longrightarrow v_v_. abc There is nothing to prove here. This holds by definition. abcliste
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Exercise:
Let T:Vlongrightarrow W be a linear map. Then: abcliste abc T is injective iff textKerT_V. abc T is surjective iff textImTW. abcliste

Solution:
Proof. abcliste abc Longrightarrow: Clearly _Vin textKerT. So _Vsubseteq T^-_W. Since T is injective T^-_W contains at most one element Longrightarrow textKerTT^-_W_V. Longleftarrow: Let v_v_in V and ase Tv_Tv_. We need to show that v_v_. Indeed we have Tv_-v_Tv_-Tv_. Longrightarrow v_-v_in textKerT. But by asption textKerT_V. Longrightarrow v_-v_ Longrightarrow v_v_. abc There is nothing to prove here. This holds by definition. abcliste
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eth, hs22, image, injective, lineare algebra, proof, vector space
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ENG (English)
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Proof
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