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https://texercises.com/exercise/isomorphism-and-bijection-1/
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Exercise:
Let T:Vlongrightarrow W be a linear map which is bijective. Then the inverse map T^-:Wlongrightarrow V is linear. In other words a linear bijective map is on isomorphism.

Solution:
proof. Let abin K w_ w_in W. Put v_T^-w_ v_T^-w_. We have Tv_w_ Tv_w_. Since T is linear we have Tav_+bv_aTv_+bTv_aw_+bw_ Longrightarrow T^-aw_+b_w_av_+bv_a T^-w_+b T^-w_ Longrightarrow T^-quad textis a linear map. For linear maps bf bijectiveiffisomorphism
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Exercise:
Let T:Vlongrightarrow W be a linear map which is bijective. Then the inverse map T^-:Wlongrightarrow V is linear. In other words a linear bijective map is on isomorphism.

Solution:
proof. Let abin K w_ w_in W. Put v_T^-w_ v_T^-w_. We have Tv_w_ Tv_w_. Since T is linear we have Tav_+bv_aTv_+bTv_aw_+bw_ Longrightarrow T^-aw_+b_w_av_+bv_a T^-w_+b T^-w_ Longrightarrow T^-quad textis a linear map. For linear maps bf bijectiveiffisomorphism
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eth, hs22, isomorphism, lineare algebra, proof, vector space
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ENG (English)
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