math185f08-hw2sol

math185f08-hw2sol - MATH 185 COMPLEX ANALYSIS FALL 2008/09...

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Unformatted text preview: MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 2 SOLUTIONS 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 . (1.1) 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 . (1.2) (a) Prove that if ϕ is C-linear, then it is R-linear. Give an example to show that the converse is false. Solution. This is obvious since R ⊂ C and so (1.1) is a special case of (1.2). For a counterexample to the converse, consider the complex conjugate function, ϕ : C → C , ϕ ( z ) = z . For λ 1 ,λ 2 ∈ R , ϕ ( λ 1 z 1 + λ 2 z 2 ) = λ 1 z 1 + λ 2 z 2 = ¯ λ 1 ¯ z 1 + ¯ λ 2 ¯ z 2 = λ 1 ¯ z 1 + λ 2 ¯ z 2 = λ 1 ϕ ( z 1 ) + λ 2 ϕ ( z 2 ) and so ϕ is R-linear. However, for λ 1 = i , z 1 = 1, λ 2 = z 2 = 0, we see that ϕ ( i ) =- i 6 = i = iϕ (1) and so it is not C-linear. (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 (1.3) for all z = x + iy ∈ C . (iii) ϕ satisfies ϕ ( z ) = ϕ (1)- iϕ ( i ) 2 z + ϕ (1) + iϕ ( i ) 2 ¯ z (1.4) for all z = x + iy ∈ C . (iv) ϕ is given by ϕ ( x + iy ) = ( ax + by ) + i ( cx + dy ) (1.5) for some a b c d ∈ R 2 × 2 . Solution. (i) ⇒ (ii): If we let λ 1 = x , z 1 = 1, λ 2 = y , z 2 = i in (1.1), we get ϕ ( z ) = ϕ ( x + yi ) = xϕ (1) + yϕ ( i ) = ϕ (1) x + ϕ ( i ) y as required. Date : October 4, 2008 (Version 1.0). 1 (ii) ⇒ (iii): Note that if z = x + iy , then x = ( z + ¯ z ) / 2 and y = ( z- ¯ z ) / 2 i . Hence ϕ ( z ) = ϕ (1) x + ϕ ( i ) y = ϕ (1) z + ¯ z 2 + ϕ ( i ) z- ¯ z 2 i = ϕ (1)- iϕ ( i ) 2 z + ϕ (1) + iϕ ( i ) 2 ¯ z. (iii) ⇒ (iv): Let a = Re ϕ (1), c = Im ϕ (1), b = Re ϕ ( i ), d = Im ϕ ( i ). Then ϕ ( x + iy ) = ϕ (1)- iϕ ( i ) 2 ( x + iy ) + ϕ (1) + iϕ ( i ) 2 ( x- iy ) = ϕ (1) x + ϕ ( i ) y = ( a + ic ) x + ( b + id ) y = ( ax + by ) + i ( cx + dy ) . (iv) ⇒ (i): With respect to the standard basis B = { 1 ,i } of C as a real vector space, (1.5) implies that ϕ has the matrix representation [ ϕ ] B , B = a b c d and is thus a R-linear function. (c) Let ϕ : C → C . Prove that the following statements are equivalent. (i) ϕ is C-linear. (ii) ϕ is R-linear and ϕ ( i ) = iϕ (1). (iii) ϕ satisfies ϕ ( z ) = ϕ (1) z (1.6) for all z ∈ C . (iv) ϕ is given by ϕ ( x + iy ) = ( ax- cy ) + i ( cx + ay ) (1.7) for some [ a c- c a ] ∈ R 2 × 2 ....
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math185f08-hw2sol - MATH 185 COMPLEX ANALYSIS FALL 2008/09...

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