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Lecture_15 - Lecture Note 15 Dr Jeff Chak-Fu WONG...

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Lecture Note 15 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff Chak-Fu WONG 1
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D OT P RODUCT IN R n D OT P RODUCT IN R n 2
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n - VECTORS In this note we shall focus on n -vectors from a geometrical point of view by generalizing the notations discussed in the preceding section. n - VECTORS 3
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As we have already seen in the first part of Lecture note 1, an n × 1 matrix u = u 1 u 2 . . . u n n × 1 , where u 1 , u 2 , . . . , u n are real numbers, which are called the components of u . Since an n -vector is an n × 1 matrix, the n -vectors u = u 1 u 2 . . . u n , v = v 1 v 2 . . . v n are said to be equal if u i = v i (1 i n ) . n - VECTORS 4
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Example 1 The 4-vectors 1 - 2 3 4 and 1 - 2 3 - 4 are not equal, since their fourth components are not the same. n - VECTORS 5
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The set of all n -vectors is denoted by R n and is called n -space . As the actual value of n need to be addressed, we regard to n -vectors simply as vectors . The real numbers are called scalars . The components of a vector are real numbers and hence the components of a vector are scalars. n - VECTORS 6
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V ECTOR O PERATIONS Definition: Let u = u 1 u 2 . . . u n , and v = v 1 v 2 . . . v n be two vectors in R n . The sum of the vectors u and v is the vector u + v = u 1 + v 1 u 2 + v 2 . . . u n + v n and it is denoted by u + v . V ECTOR O PERATIONS 7
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Example 2 If u = 1 - 2 3 and v = 2 3 - 3 are vectors in R 3 , then u + v = 1 + 2 - 2 + 3 3 + ( - 3) = 3 1 0 . V ECTOR O PERATIONS 8
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Definition: If u = u 1 u 2 . . . u n is a vector in R n and c is a scalar, then the scalar multiple c u of u by c is the vector u = c u 1 c u 2 . . . c u n . V ECTOR O PERATIONS 9
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Example 3 If u = 2 3 - 1 2 is a vector in R 4 and c = - 2 , then c u = - 2 2 3 - 1 2 = - 4 - 6 2 - 4 . V ECTOR O PERATIONS 10
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Theorem 0.1 Let u , v and w be any vectors in R n ; let c and d be any scalars. Then ( α ) u + v is a vector in R n (i.e. R n is closed under the operation of vector addition ).
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