Field and vector spaces

Field and vector spaces - Mathematics MATH 236, Winter 2008...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 9

Field and vector spaces - Mathematics MATH 236, Winter 2008...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online