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hw1 - Mathematics 185 Intro to Complex Analysis Fall 2009 M...

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Mathematics 185 – Intro to Complex Analysis Fall 2009 – M. Christ Solutions to Selecta from Problem Set 1 1.1.10(a) Let z C . Define φ z : C C by φ z ( w ) = zw . Regard φ z as a linear transfor- mation from R 2 to R 2 . Calculate its matrix representation, with respect to the standard basis of R 2 . Solution. First of all, since φ z ( w + ˜ w ) = φ z ( w ) + φ z ( ˜ w ) and φ z ( tw ) = z ( w ) for any w, ˜ w C and t R , φ z is indeed a linear transformation of R 2 , if we identify complex numbers with elements of R 2 in the standard way. I will systematically write u + iv = ( u, v ) to denote the correspondence between elements of C , and elements of R 2 . Then φ z (1 , 0) = z · (1 + i 0) = z = x + iy = ( x, y ). Similarly φ z (0 , 1) = z · (0 + i ) = iz = ix - y = ( y, x ). Thus the matrix representation of φ z , with respect to the standard basis, is x - y y x ; this is the unique matrix A which satisfies A 1 0 = x y and A 0 1 = - y x . 1.1.12(b) Using only the axioms for a field, prove that 1 z 1 + 1 z 2 = z 1 + z 2 z 1 z 2 for any two nonzero complex numbers. Solution. This is first of all an exercise in critical reading. We are asked to prove something involving 1 w for various complex numbers w . This cannot possibly be done, using only the axioms for a field — because those axioms say nothing about 1 w . We have to supplement the axioms with the notational definition 1 w = w - 1 . And we need to recognize that z 1 + z 2 z 1 z 2 means ( z 1 + z 2 )( z - 1 1 z - 1 2 ). I will also use part (a) of this problem, in the corresponding form ( z 1 z 2 ) - 1 = z - 1 1 z - 1 2 .
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