SOLUTION: Consider C as a vector space over itself. Let T : C → C be the function T(z) = ¯z. Is T a linear transformation? Justify your answer

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Question 1182571: Consider C as a vector space over itself. Let T : C → C be the function T(z) = ¯z. Is T a linear transformation? Justify your answer
Answer by ikleyn(52851) About Me  (Show Source):
You can put this solution on YOUR website!
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Although the map T transfers the sum of vectors (of complex numbers, in this case) to the sum,


the axiom of the linear transformation fails for the multiplication by the number.


Indeed, according to the axiom, it should be


    T(cz) = c*¯z,     (1)


but in reality


    ¯(cz) = ¯c*¯z,    (2)


which is DIFFERENT from  (1).


Thus the map  T   IS  NOT   a linear transformation in this case.