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mat 2310 0607_2_note1_2

# mat 2310 0607_2_note1_2 - Lecture Note 1 Dr Jeff Chak-Fu...

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Lecture Note 1: Jan 17, 2007 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310C Linear Algebra and Its Applications Spring, 2007 Produced by Jeff Chak-Fu WONG 1

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L INEAR E QUATIONS AND M ATRICES 1. Linear Systems 2. Matrices 3. Dot Product and Matrix Multiplication 4. Properties of Matrix Operations 5. Solutions of Linear Systems of Equations 6. The Inverse of A Matrix 7. Applications to Graph Theory 8. LU-Factorization L INEAR E QUATIONS AND M ATRICES 2
A PPLICATION TO G RAPH T HEORY 1. Matrix Multiplication 2. Matrix Addition A PPLICATION TO G RAPH T HEORY 3

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B ASIC CONCEPTS OF GRAPH THEORY Before we make the connection between graph theory and linear algebra, we start with some basic definitions in graph theory for those of you who are not familiar with the topic: A graph is a collection of points called vertices , joined by lines called edges . B ASIC CONCEPTS OF GRAPH THEORY 4
cf. Steven J. Leon’s book, page 52, Figure 1.3.2 B ASIC CONCEPTS OF GRAPH THEORY 5

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S O HOW IS LINEAR A LGEBRA RELATED TO GRAPH THEORY ? As you can expect, graphs can be sometimes very complicated. So one needs to find more practical ways to represent them. Matrices are a very useful way of studying graphs, since they turn the picture into numbers, and then one can use techniques from linear algebra. Given a graph G with n vertices V 1 , · · · , V n , we define the adjacency matrix of G with respect to the enumeration V 1 , · · · , V n of the vertices as being the n × n matrix A = [ a ij ] defined by a ij = 1 if { V i , V j } is an edge of the graph 0 if there is no edge joining V i and V j (1) S O HOW IS LINEAR A LGEBRA RELATED TO GRAPH THEORY ? 6
The line segment joining the vertices correspond to the edges: { V 1 , V 2 } , { V 2 , V 5 } , { V 3 , V 4 } , { V 3 , V 5 } , { V 4 , V 5 } For example, for the given graph G , the adjacency matrix (with respect to the enumeration V 1 · · · , V 5 of its vertices) is A = 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 1 1 1 0 Interpret V 1 V 2 V 3 V 4 V 5 V 1 0 1 0 0 0 V 2 1 0 0 0 1 V 3 0 0 0 1 1 V 4 0 0 1 0 1 V 5 0 1 1 1 0 S O HOW IS LINEAR A LGEBRA RELATED TO GRAPH THEORY ? 7

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A = 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 1 1 1 0 Note that the matrix A is symmetric (i.e., it is equal to its transpose). In fact, any adjacent matrix must be symmetric, for if { V i , V j } is an edge of the graph, then a ij = a ji = 1 and a ij = a ji = 0 if there is no edge joining V i and V j . In either case, a ij = a ji . S O HOW IS LINEAR A LGEBRA RELATED TO GRAPH THEORY ? 8
A WALK A walk in a graph as a sequence of edges linking one vertex to another. For example, The edges { V 1 , V 2 } , { V 2 , V 5 } represent a walk from vertex V 1 to vertex V 5 . The length of the walk is said to be 2 since it consists of two edges .

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mat 2310 0607_2_note1_2 - Lecture Note 1 Dr Jeff Chak-Fu...

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