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

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Unformatted text preview: Lecture Note 7 & 8 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong jwong@math.cuhk.edu.hk 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...

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