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Unformatted text preview: 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 z : C C by z ( w ) = zw . Regard 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 z ( w + w ) = z ( w ) + z ( w ) and z ( tw ) = t z ( w ) for any w, w C and t R , 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 z (1 , 0) = z (1 + i 0) = z = x + iy = ( x,y ). Similarly z (0 , 1) = z (0 + i ) = iz = ix- y = ( y,x ). Thus the matrix representation of z , with respect to the standard basis, is x- y y x ; this is the unique matrix A which satisfies A 1 = x y and A 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 )....
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