hw1s(1) - MATH 4377 ADVANCED LINEAR ALGEBRA SUMMER 2011 Key...

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Unformatted text preview: MATH 4377, ADVANCED LINEAR ALGEBRA, SUMMER 2011, Key to HW#1 Due date: Tuesday, June 7th Remark : In order to do this set of hw, you may need to read the Appendics about the definition of the fields. 1. Recall that complex numbers are of the form x = a + bi with a,b are (arbitrary) real numbers and i = √- 1 (i.e. i 2 =- 1). Let C be the set of complex numbers. (a) Show that C (with the standard addition and multiplication) is field by verifying F1-F5 on Page 553. (b) Let x = 3 + 5 i . Find x- 1 . Solution . (a) F1-F5 can easily be verified. (b) (3 + 5 i )- 1 = 1 3 + 5 i = 3- 5 i (3 + 5 i )(3- 5 i ) = 3- 5 i 9 + 25 = 3 34- 5 34 i. 2. Let S = { a,b } . Define the addition “+” and multiplication “ · ” on S by the following charts: + a b a a b b b a · a b a a a b a b (1) Prove that S a field under the the addition “+” and multiplication “ · ” defined above by verifying F1-F5 on Page 553....
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hw1s(1) - MATH 4377 ADVANCED LINEAR ALGEBRA SUMMER 2011 Key...

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