# verkefnablad02_lausnir - Línuleg algebra A og B(STÆ107G...

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Unformatted text preview: Línuleg algebra A og B (STÆ107G, STÆ106G, HAG105G) Verkefnablað 2 Lausnir 7.9.2011 1. Reiknið eftirfarandi. (a) (8 + 6 i ) + (3 + 2 i ) . (b) (4 + 5 i )- (6- 3 i ) . Lausn: (a) (8 + 6 i ) + (3 + 2 i ) = 8 + 3 + 6 i + 2 i = 11 + (6 + 2) i = 11 + 8 i . (b) (4 + 5 i )- (6- 3 i ) = 4- 6 + 5 i + 3 i =- 2 + (5 + 3) i =- 2 + 8 i . 2. Reiknið eftirfarandi. (a) √- 16 · √- 25 . (b) (1 + 2 i )(1 + 3 i ) . (c) (3- 7 i ) 2 . (d) i 37 . Lausn: (a) √- 16 √- 25 = √- 1 · 16 √- 1 · 25 = √- 1 √ 16 √- 1 √ 25 = i · 4 · i · 5 = 20 i 2 =- 20 . (b) (1 + 2 i )(1 + 3 i ) = 1 + 3 i + 2 i + 6 i 2 = 1- 6 + (3 + 2) i =- 5 + 5 i . (c) (3- 7 i ) 2 = 3 2- 2 · 3 · 7 i + (7 i ) 2 = 9- 42 i + 49 i 2 = 9- 49 + 42 i =- 40 + 42 i . (d) i 37 = i 36 · i = ( i 2 ) 18 · i = (- 1) 18 · i = 1 · i = i . 3. Reiknið eftirfarandi. (a) (5 + 7 i )(3- 2 i ) . (b) (8 i )(1- i ) . Lausn: (a) (5 + 7 i )(3- 2 i ) = 15- 10 i + 21 i- 14 i 2 = 15 + 14 + (21- 10) i = 29 + 11 i . (b) (8 i )(1- i ) = 8 i- 8 i 2 = 8 + 8 i . 4. Reiknið eftirfarandi. (a) 2- 5 i 1- 6 i (b) 1 + 5 i 4 + 3 i Lausn: (a) 2- 5 i 1- 6 i = 2- 5 i 1- 6 i · 1 + 6 i 1 + 6 i = (2- 5 i )(1 + 6 i ) (1- 6 i )(1 + 6 i ) = 2 + 12 i- 5 i- 30 i 2 1 2- (6 i ) 2 = 2 + 30 + (12- 5) i 1- 36 i 2 = 32 + 7 i 1 + 36 = 32 37 + 7 37 i....
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verkefnablad02_lausnir - Línuleg algebra A og B(STÆ107G...

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