Lecture_7

# Lecture_7 - Lecture Note 7& 8 Dr Jeff Chak-Fu WONG...

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Unformatted text preview: Lecture Note 7 & 8 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 R EAL V ECTOR S PACES 1. Vector Spaces 2. Subspaces 3. Linear Independence 4. Basis and Dimension 5. Coordinates and Change of Basis 6. Homogeneous Systems 7. The Rank of a Matrix and Applications REAL VECTOR SPACES 2 V ECTORS IN R n 1. n-vectors VECTORS IN R n 3 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 As we have already seen in the first part of Lecture 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 5 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 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 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 . VECTOR OPERATIONS 8 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     . VECTOR OPERATIONS 9 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         . VECTOR OPERATIONS 10 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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Lecture_7 - Lecture Note 7& 8 Dr Jeff Chak-Fu WONG...

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