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Unformatted text preview: AXIOMS FOR VECTOR SPACES MATH 108A, March 28, 2010 Among the most basic structures of algebra are fields and vector spaces over fields. It is well worth the effort to memorize the axioms that define fields and vector spaces. 1 Field axioms Definition. A field is a set F together with two operations (functions) f : F F F, f ( x,y ) = x + y and g : F F F, g ( x,y ) = xy, which satisfy the following axioms: 1. addition is commutative: x + y = y + x , for all x,y F . 2. addition is associative: ( x + y ) + z = x + ( y + z ), for all x,y,z F . 3. existence of additive identity: there is an element 0 F such that x +0 = x , for all x F . 4. existence of additive inverses: if x F , there is an element- x F such that x + (- x ) = 0. 5. multiplication is commutative: xy = yx , for all x,y F . 6. multiplication is associative: ( xy ) z = x ( yz ), for all x,y,z F . 7. existence of multliplicative identity: there is an element 1 F such that 1 6 = 0 and x 1 = x , for all x F . 8. existence of multliplicative inverses: if x F and x 6 = 0, there is an element (1 /x ) F such that x (1 /x ) = 1. 9. distributivity: x ( y + z ) = xy + xz , for all x,y,z F . 1 Example 1. Recall that a rational number is simply the ration of two integers....
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