LA07_6 - Linear Algebra Spring 2007 Chapter 6 Vector spaces Generalization of Euclidean vectors we want to construct an arbitrary vector space by

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___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University - Ansan 6-1 Chapter 6. Vector spaces Generalization of Euclidean vectors; we want to construct an arbitrary vector space, by defining some basic properties to be satisfied. 1. Vector spaces Definition Let V be a set on which two operations, called addition and scalar multiplication , have been defined. If two vectors u and v are in V , the sum of u and v is denoted by u + v , and if c is a scalar, the scalar multiple of u by c is denoted by c u . If the following axioms hold for all u , v , and w in V and for all scalars c and d , then V is called a vector space and its elements are called vectors; 1. u + v in V ( closed under addition ) 2. u + v = v + u ( commutative rule ) 3. ( u + v )+ w = u +( v + w ) ( associative rule ) 4. There exists an element 0 in V , called a zero vector , s.t. u + 0 = u . ( additive identity ) 5. For each u in V , there is an element – u in V s.t. u +(- u )= 0 . ( additive inverse ) 6. c u in V ( closed under scalar multiplication ) 7. c ( u + v )= c u + c v ( distributive rule ) 8. ( c+d) u = c u + d u ( distributive rule ) 9. c ( d u )=( cd) u 10. 1 u = u By scalars, we mean the real numbers. Accordingly, we should refer V as a vector space over the real numbers. In fact, the scalars can be any number system (known as a field ), in which we can add, subtract, multiply, and divide according to the usual laws of arithmetic. Roughly, a vector space consists of a set of elements, a field, and vector operations. Note the vector operations may be different from our usual arithmetic. Two most important properties of a vector space is 1 and 6 ( closure ). For example, (1) For any 1 n , n is a vector space (so called Euclidean vector space) with usual operations of addition and scalar multiplication. (2) The set of all n m × matrices is a vector space with the usual operations of matrix addition and scalar multiplication. This vector space is denoted M mn . (3) Let P 2 denote the set of all polynomials of degree 2 or less with real coefficients; i.e. { } + + = 2 1 0 2 2 1 0 2 , , ; a a a x a x a a P . We define addition and scalar multiplication in the usual way. If 2 2 1 0 ) ( x a x a a x p + + = and 2 2 1 0 ) ( x b x b b x q + + = , then
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___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University - Ansan 6-2 2 2 2 1 1 0 0 ) ( ) ( ) ( ) ( ) ( x b a x b a b a x q x p + + + + + = + 2 2 1 0 ) ( x ca x ca ca x cp + + = (4) Let F [ a,b ] denote the set of all real valued functions defined on the closed interval [ a,b ]. If f and g are two such functions and c is a scalar, then f+g and cf are defined by ) ( ) ( ) )( ( x g x f x g f + = + and ) ( ) )( ( x
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This note was uploaded on 06/06/2010 for the course MATH 54 taught by Professor Chung during the Spring '07 term at 카이스트, 한국과학기술원.

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LA07_6 - Linear Algebra Spring 2007 Chapter 6 Vector spaces Generalization of Euclidean vectors we want to construct an arbitrary vector space by

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