This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 )....
View
Full
Document
 Spring '07
 Lim
 Math

Click to edit the document details