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Lecture_8_1

# Lecture_8_1 - Lecture Note 8-1 Oct 31 Nov 3 2006 Dr Jeff...

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Lecture Note 8-1: Oct 31 - Nov 3, 2006 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected]k.edu.hk MAT 2310 Linear Algebra and Its Applications Fall, 2006 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
R OW S PACE OF A M ATRIX R OW S PACE OF A M ATRIX 3

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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 R OW S PACE OF A M ATRIX 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 0 1 3 1 0 2 0 1 3 B = 1 - 1 - 2 3 - 2 - 3 1 - 1 - 2 0 1 3 1 0 1 0 1 3 C = 1 - 1 - 1 4 - 3 - 1 3 - 1 3 1 - 1 - 1 0 1 3 0 2 6 1 - 1 - 1 0 1 3 0 0 0 1 0 2 0 1 3 0 0 0 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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Lecture_8_1 - Lecture Note 8-1 Oct 31 Nov 3 2006 Dr Jeff...

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