# Lecture1 - Lecture 1 Introduction of General Vector Spaces...

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1 Lecture 1 Introduction of General Vector Spaces Recall: A set V in Euclidean space R n is closed under two operations “addition” and “scalar multiplication” and satisfy Properties of vector addition: A1: A2: A3: A4: Properties of vector scalar multiplication: S1: S2: S3: S4: 1.1 Field: A field is a set F equipped with two binary operations “addition” and “multiplication” satisfy the following properties: addition: AA1: a + ( b + c ) = ( a + b ) + c AA2: a + b = b + a AA3: 0, s.t. a + 0 = a AA4: a , s.t. a + (− a ) = 0 multiplication: SS1: a ( b c ) = ( a b ) c SS2: a b = b a SS3: 1, s.t. a 1 = a . SS4: a −1 , s.t. a a −1 = 1 for a ≠ 0 Moreover, a ( b + c ) = ( a b ) + ( a c ).

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2 Example: The set of all rational numbers Q = { a / b | a , b in Z , b ≠ 0 } where Z is the set of integers. Example: Z = { 0, 1, 2, …}, set of integers. 1.2 Finite field
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## This note was uploaded on 12/25/2010 for the course MATHEMATIC MATB24 taught by Professor Yang during the Fall '09 term at University of Toronto.

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Lecture1 - Lecture 1 Introduction of General Vector Spaces...

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