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lina-23-10-09-p1 - z 1 6 = 0 then 1/z 1 ∈ Q √ 2(hint...

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Linear Algebra & Geometry: Sheet 3 Set on Friday, October 23: Questions 1, 2, 3,4 1. Use the relation e i ϕ = cos ϕ + i sin ϕ to prove De Moivre’ s Theorem : for any n N cos( ) + i sin( ) = (cos ϕ + i sin ϕ ) n Use this formula to derive the following relations cos(3 ϕ ) = 4 cos 2 ϕ - 3 cos ϕ sin(3 ϕ ) = - 4 sin 3 ϕ + 3 sin ϕ . 2. The following extension of the rational numbers is analogous to the construction of the complex numbers from the real numbers. Consider numbers of the form z = x + 2 y where x and y are rational numbers. We call the set of all these numbers Q ( 2), i.e., Q ( 2) = { x + 2 y ; x, y Q } . Show that if z 1 , z 2 Q ( 2) then (i) z 1 + z 2 Q ( 2) (ii)
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Unformatted text preview: z 1 6 = 0 then 1 /z 1 ∈ Q ( √ 2) (hint: use the fact that √ 2 is irrational.) (iv) If z 1 6 = 0 then z 2 /z 1 ∈ Q ( √ 2) 3. Compute the following dot products and determine the cosine of the angle between the vectors. ( a ) ± 1 2 ² · ± 6-8 ² ( b ) ±-7-3 ² · ± 1 ² 4. Use the relation between the norm and the dot-product, k v k 2 = v · v , to show (i) the parellelogram law: k v-w k 2 + k v + w k 2 = 2 k v k 2 + 2 k w k 2 (ii) the polarisation identity: v · w = 1 4 ( k v + w k 2- k v-w k 2 ) 1...
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