185hw1.pdf - Math 185 Homework 1 \u2013 Due 09\/04 Peter Koroteev 1 Show that the system of all matrices of the form \u0012 \u0013 \u03b1 \u03b2 \u2212\u03b2 \u03b1 combined by matrix

# 185hw1.pdf - Math 185 Homework 1 – Due 09/04 Peter...

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Math 185 Homework 1 – Due 09/04 Peter Koroteev 1. Show that the system of all matrices of the form α β - β α combined by matrix addition and matrix multiplication, is isomorphic to the field of complex numbers. 2. Show that the equation | z | 2 - 2Re(¯ az )+ | a | 2 = ρ 2 represents a circle of radius ρ centered at a . 3. For fixed a C show that | z - a | | 1 - ¯ az | = 1 , if | z | = 1 and 1 - ¯ az 6 = 0. 4. Prove that 1 + cos θ + cos 2 θ + · · · + cos = 1 2 + sin ( ( n + 1 2 ) θ ) 2 sin #### You've reached the end of your free preview.

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Unformatted text preview: θ2 , whereθ is a real angle. 5. Fix n≥ 1. Show that the n th roots of unity ω , . . . , ω n- 1satisfy (a)( z-ω )(z- ω 1) · · · (z- ω n-1 ) = zn- 1, (b) n- 1 Xa =0ω a= 0 , (c) n-1 Ya =0ω a= (- 1) n-1 ,(d) n- 1X a =0 ω k a= (, 1≤ k≤ n-1 n ,k =n .6. Find formulae for cos 4θ and sin 4 θ in terms of cos θ and sin θ ....
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