The given function-
T(⎣⎡xyz⎦⎤)=⎣⎡x+y−z2xyx+z+1⎦⎤
Letα=⎣⎡α1α2α3⎦⎤,β=⎣⎡β1β2β3⎦⎤∈R3
T(α+β)=T(⎣⎡α1α2α3⎦⎤+⎣⎡β1β2β3⎦⎤)=T(⎣⎡α1+β1α2+β2α3+β3⎦⎤)=⎣⎡α1+β1+α2+β2−(α3+β3)2(α1+β1)(α2+β2)α1+β1+α3+β3+1⎦⎤=⎣⎡α1+α2−α32α1α2α1+α3+1⎦⎤+⎣⎡β1+β2−β32β1β2+2(α1β2+β1α2)β1+β3⎦⎤=T(α)+T(β)
Since, T(β)=⎣⎡α1+α2−α32α1alpha2α1+alpha3+1⎦⎤=⎣⎡β1+β2−β32β1β2+2(α1β2+β1α2)β1+β3⎦⎤
Therefore T is not liner transformation between vector space.
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