math185f08-hw2 - MATH 185: COMPLEX ANALYSIS FALL 2008/09...

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 2 1. Recall that C is both a real vector space of dimension 2 and a complex vector space of dimen- sion 1. A function : C C is called R-linear if is a linear transformation of real vector spaces, ie. ( 1 z 1 + 2 z 2 ) = 1 ( z 1 ) + 2 ( z 2 ) for all 1 , 2 R and z 1 ,z 2 C . It is called C-linear if is a linear transformation of complex vector spaces, ie. ( 1 z 1 + 2 z 2 ) = 1 ( z 1 ) + 2 ( z 2 ) for all 1 , 2 C and z 1 ,z 2 C . (a) Prove that if is C-linear, then it is R-linear. Give an example to show that the converse is false. (b) Let : C C . Prove that the following statements are equivalent. (i) is R-linear. (ii) satisfies ( z ) = (1) x + ( i ) y for all z = x + iy C . (iii) satisfies ( z ) = (1)- i ( i ) 2 z + (1) + i ( i ) 2 z for all z = x + iy C ....
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math185f08-hw2 - MATH 185: COMPLEX ANALYSIS FALL 2008/09...

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