MATH 301 Homework 3 - ² : (b) Show that C with the...

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Math 301, Homework 3 1. Let K be the set of all 2 2 matrices of the form a ± b b a ± with a; b real. Consider the map C ! K z ) = x ± y y x ± if z = x + iy (a) Show that is a bijection that preserves addition and multiplication. (i.e. z 1 z 2 z 1 + z 2 ) and z 1 z 2 z 1 z 2 ) ) Remark: In this problem you have shown that the set K together di/erent is essentially the same as C : (b) What are the zero, identity and the multiplicative inverse of a given matrix in K ? 2. In C z 1 = x 1 + iy 1 ; z 2 = x 2 + iy 2 with x i ; y i z 1 ² z 2 if x 1 < x 2 ( y 1 ; y 2 may be anything) or x 1 = x 2 and y 1 ³ y 2 : (a) Compare z 1 = 3 ± 2 i; z 2 = i; z 3 = 3 + 2 i; z 4 = 1 + 2 i ; one is larger than which according to
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Unformatted text preview: ² : (b) Show that C with the relation ² satis&amp;es all the order axioms except one. Which axiom does it not satisfy? (For the axiom it will not satisfy do not try a general proof, it is enough to give a counter example.) 3. Using the order &amp;eld axioms, prove the following: (a) If x ³ y and &lt; x; then y &amp; 1 ³ x &amp; 1 : (b) ( ± 1) ´ x = ± x 4. If F is an ordered &amp;eld, we showed in class that it has a subset Q F that looks like Q consisting of all elements of the form p F ´ q &amp; 1 F where p; q are integers, q 6 = 0 and n F = 1 F +1 F + ::: +1 F ( n times) for a positive integer n: Use the order axioms and other results we proved in class to show that 1 F ³ 5 F ´ 3 &amp; 1 F ³ 2 F : 1...
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This note was uploaded on 06/25/2010 for the course MATH MATH 301 taught by Professor Alberterkip during the Fall '08 term at Sabancı University.

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