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Unformatted text preview: Lecture Note 1: Jan 15, 2007 Dr. Jeff ChakFu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310C Linear Algebra and Its Applications Spring, 2007 Produced by Jeff ChakFu WONG 1 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. LUFactorization LINEAR EQUATIONS AND MATRICES 2 P ROPERTIES OF M ATRIX O PERATIONS 1. Properties of Matrix Addition 2. Properties of Matrix Multiplication • Definition of Identity Matrix • Power of A Matrix 3. Properties of Scalar Multiplication 4. Properties of Transpose • Definition of Symmetric Matrix PROPERTIES OF MATRIX OPERATIONS 3 D EFINITION OF S YMMETRIC M ATRIX DEFINITION OF SYMMETRIC MATRIX  A matrix A = [ a ij ] with real entries is called symmetric if A T = A. That is, A is symmetric if it is a square matrix for which a ij = a ji (Exercise). If matrix A is symmetric, then the elements of A are symmetric with respect to the main diagonal A . DEFINITION OF SYMMETRIC MATRIX 4 Example 1 The matrices A = 1 2 3 2 4 5 3 5 6 and I 3 = 1 0 0 0 1 0 0 0 1 are symmetric. DEFINITION OF SYMMETRIC MATRIX 5 S OLUTIONS OF L INEAR S YSTEMS OF E QUATIONS SOLUTIONS OF LINEAR SYSTEMS OF EQUATIONS 6 Our ultimate goal is to study: 1. Solving linear system (a) GaussJordan reduction reduced row echelon method + elementary row operations (b) Gaussian elimination row echelon method + elementary row operations + back substitution (c) Consistent/Inconsisent system i. If there is at least one solution, it is called consistent . ii. A system of equations that has no solutions is said to be inconsistent . 2. Homogeneous system (a) trivial solution (b) nontrivial solution SOLUTIONS OF LINEAR SYSTEMS OF EQUATIONS 7 DEFINITION  An m × n matrix A is said to be in reduced row echelon form if it satisfies the following properties: (a) All zero rows, if there are any, appear at the bottom of the matrix. (b) The first nonzero entry from the left of a nonzero row is a 1 . This entry is called a leading one of its row. (c) For each nonzero row, the leading one appears to the right and below any leading one’s in preceding rows. (d) If a column contains a leading one, then all other entries in that column are zero. Note that a matrix in reduced row echelon form appears as a staircase (“echelon") pattern of leading ones descending from the upper left corner of the matrix. An m × n matrix satisfying properties (a) , (b) and (c) is said to be in row echelon form . Example 2 The following are matrices in reduced row echelon form since they satisfy properties (a), (b), (c) and (d): A = 1 1 1 1 , B = 1 0 0 0 2 4 0 1 0 0 4 8 0 0 0 1 7 2 0 0 0 0 0 0 0 0 ...
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 Spring '11
 JeffWong
 Linear Algebra, Algebra

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