math185f08-hw2

# 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 ϕ : C → C is called R-linear if ϕ is a linear transformation of real vector spaces, ie. ϕ ( λ 1 z 1 + λ 2 z 2 ) = λ 1 ϕ ( z 1 ) + λ 2 ϕ ( z 2 ) for all λ 1 ,λ 2 ∈ R and z 1 ,z 2 ∈ C . It is called C-linear if ϕ is a linear transformation of complex vector spaces, ie. ϕ ( λ 1 z 1 + λ 2 z 2 ) = λ 1 ϕ ( z 1 ) + λ 2 ϕ ( z 2 ) for all λ 1 ,λ 2 ∈ C and z 1 ,z 2 ∈ C . (a) Prove that if ϕ is C-linear, then it is R-linear. Give an example to show that the converse is false. (b) Let ϕ : C → C . Prove that the following statements are equivalent. (i) ϕ is R-linear. (ii) ϕ satisfies ϕ ( z ) = ϕ (1) x + ϕ ( i ) y for all z = x + iy ∈ C . (iii) ϕ satisfies ϕ ( z ) = ϕ (1)- iϕ ( i ) 2 z + ϕ (1) + iϕ ( 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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