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Lecture_4_1

# Lecture_4_1 - Lecture Note 5 Oct 3 Oct 7 2006 Dr Jeff...

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Lecture Note 5: Oct 3 - Oct 7, 2006 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2006 Produced by Jeff Chak-Fu WONG 1

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R EAL V ECTOR S PACES 1. Vector Spaces 2. Subspaces 3. Linear Independence 4. Basis and Dimension 5. Homogeneous Systems 6. The Rank of a Matrix and Applications 7. Coordinates and Change of Basis 8. Orthonormal Bases in R n 9. Orthogonal Complements R EAL V ECTOR S PACES 2
V ECTOR S PACES V ECTOR S PACES 3

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We have already defined R n and examined some of its basic properties in Theorem 0.1 of Lecture Note 3-1. Our aim is to study the notion of a vector space. In particular, we shall study the properties and structure of a vector space . V ECTOR S PACES 4
Definition 1: A real vector space is a set of elements V together with two operations and fl satisfying the following properties: ( α ) If u and v are any of elements of V , then u v is in V (i.e., V is closed under the operation ). ( A 1 ) u v = v u , for u and v in V (commutative property). ( A 2 ) u ( v w ) = ( u v ) u , for u , v and w in V (associate property). ( A 3 ) There is an element 0 (called zero vector) in V such that u 0 = 0 u = u , for all u in V. ( A 4 ) For each u in V , there is an element - u (called the negative of u ) in V such that u - u = 0 . ( β ) If u is any element of V and c is any real number, then c fl u is in V (i.e., V is closed under the operation fl ). ( M 1 ) c fl ( u v ) = c fl u c fl v , for all real numbers c and all u and v in V . ( M 2 ) ( c + d ) fl u = c fl u d fl u , for all real numbers c and d and all u in V . ( M 3 ) c fl d fl u = ( cd ) fl u , for all real numbers c and d and all u in V . ( M 4 ) 1 fl u = u , for all u in V . V ECTOR S PACES 5

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The element of V are called vectors ; the real numbers are called scalars . The operation is called vector addition ; the operation fl is called scalar multiplication . 1. (a) The vector 0 in property ( A 3 ) is called a zero vector . (b) The vector - u in property ( A 4 ) is called a negative of u . (c) It can be shown (cf. Page 30-33) that the vector 0 and - u are unique. 2. (a) Property ( α ) is called the closure property for . (b) Property ( β ) is called the closure property for fl . (c) We also say that V is closed under the operations of vector addition , and scalar multiplication , fl . V ECTOR S PACES 6
Example 1 Consider the set R n together with the operation of vector addition and scalar multiplication as defined in Lecture note 3-1. Theorem 0.2 in Lecture note 3-1 (cf. Page 62) established the fact that R n is a vector space under the operations of addition and scalar multiplication of n -vectors. V ECTOR S PACES 7

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Example 2 Consider the set V of all ordered triples of real numbers of the form ( x, y, 0) and define the operations and fl by ( x, y, 0) ( x 0 , y 0 , 0) = ( x + x 0 , y + y 0 , 0) c fl ( x, y, 0) = ( cx, cy, 0) .
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