Question: Consider the function f : C - M2x2(R) defined by f (a + bi) = ] a a (a) Prove that f is linear over

Consider the function f : C - M2x2(R) defined by f (a + bi) = ] a a (a) Prove that f is linear over R by proving that: ( i) f ( z + w) = f(z) + f (w) for all z, w E C. (ii) f(tz) = tf(z) for all z E C and t E R. (Pay close attention to the fact that t ( R!) (b) Prove that f is multiplicative by proving that: f(zw) = f(z)f(w) for all z, w E C. (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.]

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