lina-16-10-09-p1

# lina-16-10-09-p1 - z = x i y we deﬁned ¯ z:= x-i y and |...

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Linear Algebra & Geometry: Sheet 2 Set on Friday, October 16: Questions 1, 2, 4 (a)(c)(e), 5 (i), 6 1. Consider the following vectors: ( a ) 1 1 ( b ) 0 - 5 ( c ) - 3 4 , find for each of them a λ > 0 and a θ [0 , 2 π ) such that v = λ u ( θ ) where u ( θ ) = cos θ sin θ . 2. Find the components of the vector x R 2 that satisfies x = λ u ( θ ) for (a) λ = 1 and θ = π/ 3 (b) λ = 2 and θ = π/ 2 (c) λ = 10 and θ = 7 π 6 3. Let v 1 , v 2 , · · · , v n R 2 be n arbitrary vectors in R 2 . Show that k v 1 + v 2 + · · · + v n k ≤ k v 1 k + k v 2 k + · · · + k v n k and give an example of n vectors for which there is equality. 4. Compute the following expressions, i.e., write them in the form a + i b with explicit numbers a and b , ( a ) (1 + i) + (2 - 3i) ( b ) (1 + i)(1 + 2i) - 3 ( c ) 1 1 - i ( d ) 2 + i 1 + i ( e ) (2 - 3i)(3 - 2i) 1 + i ( f ) 1 (1 + i)(2 - i) 5. Recall that for a complex number
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Unformatted text preview: z = x + i y we deﬁned ¯ z := x-i y and | z | = √ ¯ zz . Show that (i) ( a ) z 1 + z 2 = ¯ z 1 + ¯ z 2 ( b ) z 1 z 2 = ¯ z 1 ¯ z 2 ( c ) z 1 /z 2 = ¯ z 1 / ¯ z 2 ( d ) ¯ ¯ z = z (ii) ( e ) | z 1 z 2 | = | z 1 || z 2 | ( f ) | z 1 + z 2 | ≤ | z 1 | + | z 2 | 6. (i) Find x,y ∈ R such that ( a ) e i π 3 = x + i y ( b ) 2e i π 2 = x + i y ( c ) 10e i 7 π 6 = x + i y and compare with Question 2 . (ii) Find r ∈ R + , ϕ ∈ [0 , 2 π ) such that ( a ) 1 + i = r e i ϕ ( b )-5i = r e i ϕ ( b )-3 + 4i = r e i ϕ and compare with Question 1 . 1...
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