farkhsLv1

11 Mar 2023

Q3. Consider the function f: C    defined by

 

 

(a) Prove that f is linear over  by proving that:

(i) f (z+w) = f(z) + f(w) for all z, w .

(ii) f (t z) = t f(z) for all z  and t  . (Pay close attention to the fact that t  !)

(b) Prove that f is multiplicative by proving that:

f (z w) = f(z) f(w) for all z, w .

 

(c) Prove that f is one-to-one. [Warning: Since f is not a linear transformation as per our definition in Chapter 5, you shouldn't apply any of the one-to-one criteria given there. However, the definition of one-to-one given in Chapter 5 can be applied to any function. That is, f is one-to-one if whenever f (z) = f(w) then z = w.]

[Note: This problem shows that we can "identify" C with the set of 2 x 2 real matrices of the form, in the sense that every complex number corresponds uniquely to such a matrix, and moreover the addition and multiplication of complex numbers matches the addition and multiplication of these corresponding matrices! This is an example of an isomorphism.]