mat 2310 0607_2_note3

mat 2310 0607_2_note3 - Lecture Note 3 Jan 29 31 2007 Dr...

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Lecture Note 3: Jan 29 & 31, 2007 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310C Linear Algebra and Its Applications Spring, 2007 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
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V ECTORS IN R n 1. n -vectors V ECTORS IN R n 3
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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 4
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As we have already seen in the first part of Lecture 0, 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 5
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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 6
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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 7
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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 8
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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 9
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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 10
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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 .
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