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Unformatted text preview: Mathematics MATH 236, Winter 2008 Algebra 2 Fields and vector spaces/ definitions and examples Most of linear algebra takes place in structures called vector spaces. It takes place over structures called fields, which we now define. DEFINITION 1 . A field is a set (often denoted F ) which has two binary operations + F ( addition ) and F ( multiplication ) defined on it. (So for any a,b F , a + F b and a F b are elements of F .) They must satisfy the following rules: 1. There is an element 0 F F such that 0 F + F a = a for any a F . (0 F is the additive identity , or just the zero of the field.) 2. For any element a F , there is an element- a F such that (- a )+ F a = F . (- a is the additive inverse or the negative of a .) 3. For any a,b,c F , a + F ( b + F c ) = ( a + F b ) + F c . (This is the associative law for addition.) 4. For any a,b F , a + F b = b + F a (the commutative law of addition). 5. There is an element 1 F F with 1 F 6 = 0 F such that 1 F a = a for any a F . (1 F is the multiplicative identity , or just the identity or (sometimes) unity of the field.) 6. For any element 0 F 6 = a F , there is an element a- 1 F such that ( a- 1 ) F a = 1 F . ( a- 1 is the multiplicative inverse or the inverse of a .) 7. For any a,b,c F , a F ( b F c ) = ( a F b ) F c . (This is the associative law for multiplication.) 8. For any a,b F , a F b = b F a (the commutative law of multiplication). 9. For any a,b,c F , a F ( b + F c ) = ( a F b )+ F ( a F c ) (the distributive law ). REMARKS. 1. All of these rules are true for the reals (we will denote the set of reals, and the field, by R ), the complex numbers (denoted C ), and the rationals (denoted Q ). So each of these is an example of a field. (This is with the usual, familiar, operations.) I will list some other, less familiar, examples below. 2. Contrarily, the integers (positive, negative and zero) we will use Z for the integers with the usual operations, dont make a field. The only problem is the multiplicative inverse while any nonzero integer has a multiplicative inverse (in Q ), it isnt always in Z , e.g., 1 2 is not an integer. ( Z is not closed under multiplicative inverses , to use the technical term.) 1 3. In this definition, I should emphasize that the elements 0 F and 1 F may not be the usual 0 and 1. Similarly, even if there is a standard + and , it is possible that + F and F are different operations. (For instance, its quite possible to define unusual operations + Z and Z on the integers that make them into a field. I can do it, but its messy.) 4. The exception for 0 F in the article regarding multiplicative inverses is necessary; its not hard to prove from the other rules that b F = 0 F for any b F . (See below.) 5. Terminology alert! The first four axioms dont say anything about mul- tiplication. They state that F is a commutative group (a.k.a. an(a....
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