This preview shows pages 1–12. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture Note 7 & 8 Dr. Jeff ChakFu 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 ChakFu 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. nvectors VECTORS IN R n 3 n VECTORS In this note we shall focus on nvectors from a geometrical point of view by generalizing the notations discussed in the preceding section. nVECTORS 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 nvector is an n × 1 matrix, the nvectors 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 ) . nVECTORS 5 Example 1 The 4vectors 1 2 3 4 and 1 2 3 4 are not equal, since their fourth components are not the same. nVECTORS 6 The set of all nvectors is denoted by R n and is called nspace . As the actual value of n need to be addressed, we regard to nvectors 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. nVECTORS 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 ....
View
Full
Document
 Spring '06
 Dr.JeffChakFuWONG
 Algebra, Vector Space

Click to edit the document details