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https://texercises.com/exercise/kernel-and-image/
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Exercise:
Let VW be vector spaces over K and T:Vlongrightarrow W a linear map. abcliste abc The set textKerT:vin V| Tv_Wsubseteq V is a linear subspace of V. It is called the kernel of T. abc textImT:Tv|vin Vsubseteq W is a linear subspace of W. It is called the image of T. abcliste

Solution:
Proof. abcliste abc Let abin K v_v_in textKerT. Longrightarrow Tav_+bv_aTv_+bTv_a +b Longrightarrow av_+bv_extKerT. Also _VextKerT because T_V_W. Longrightarrow textKerTsubseteq V is a subspace abc Let abin K w_ w_in textImT. Longrightarrow exists v_v_in V s.t. Tv_w_ Tv_w_. Now Tav_+bv_aTv_+bTv_aw_+bw_. Longrightarrow aw_+bw_extImT. Also _Win textImT because T_V_W. Longrightarrow textImT is a subspace of W. abcliste
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Exercise:
Let VW be vector spaces over K and T:Vlongrightarrow W a linear map. abcliste abc The set textKerT:vin V| Tv_Wsubseteq V is a linear subspace of V. It is called the kernel of T. abc textImT:Tv|vin Vsubseteq W is a linear subspace of W. It is called the image of T. abcliste

Solution:
Proof. abcliste abc Let abin K v_v_in textKerT. Longrightarrow Tav_+bv_aTv_+bTv_a +b Longrightarrow av_+bv_extKerT. Also _VextKerT because T_V_W. Longrightarrow textKerTsubseteq V is a subspace abc Let abin K w_ w_in textImT. Longrightarrow exists v_v_in V s.t. Tv_w_ Tv_w_. Now Tav_+bv_aTv_+bTv_aw_+bw_. Longrightarrow aw_+bw_extImT. Also _Win textImT because T_V_W. Longrightarrow textImT is a subspace of W. abcliste
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eth, hs22, image, lineare algebra, proof
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ENG (English)
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