Exercise 3. Let T : C → C be the conjugate function z + ž, that sends a complex number z = a + bi to its conjugate z = a – bi. Let us consider C as a vector space over R and fix the basis B = {1,i}. (i) Show that T is R-linear. (ii) Compute T(1) and T(i). Write them as linear combinations of the elements of the basis {1+i, 1 – i}. (iii) Let S: C → C be the map z → 2z. Compute S(1) and S(i). (iv) Recall that S o T is the linear application obtained by composition. What are the values of So T(1) and S o T (i)?

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