Lecture_8_1 - Lecture Note 8-1: Oct 31 - Nov 3, 2006 Dr....

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Unformatted text preview: Lecture Note 8-1: Oct 31 - Nov 3, 2006 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, 2006 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. 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 REAL VECTOR SPACES 2 R OW S PACE OF A M ATRIX ROW SPACE OF A MATRIX 3 Example 1 Determine whether the following matrices have the same row space: A = 1 1 5 2 3 13 , B = 1- 1- 2 3- 2- 3 , C = 1- 1- 1 4- 3- 1 3- 1 3 ROW SPACE OF A MATRIX 4 Solution Matrices have the same row space if and only if their reduced row echelon forms have the same nonzero rows; A = 1 1 5 2 3 13 → 1 1 5 1 3 → 1 2 1 3 B = 1- 1- 2 3- 2- 3 → 1- 1- 2 1 3 → 1 1 1 3 C = 1- 1- 1 4- 3- 1 3- 1 3 → 1- 1- 1 1 3 2 6 → 1- 1- 1 1 3 → 1 2 1 3 Since the nonzero rows of the reduced row echelon form of A and of the reduced row echelon form of C are the same , A and C have the same row space. On the other hand, the nonzero rows of the reduced row echelon form of B are not the same as the others, and so B has a different row space....
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This note was uploaded on 10/15/2009 for the course MATHEMATIC MAT2310B taught by Professor Jeffwong during the Spring '09 term at CUHK.

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Lecture_8_1 - Lecture Note 8-1: Oct 31 - Nov 3, 2006 Dr....

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