Lecture_1_1 - Lecture Note 1: Sept 11 - 15, 2006 Dr. Jeff...

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Unformatted text preview: Lecture Note 1: Sept 11 - 15, 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 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. LU-Factorization 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 P ROPERTIES OF M ATRIX A DDITION Theorem 0.1 (Properties of Matrix Addition) Let A,B,C and D be m × n matrices. (a) A + B = B + A (Commutative law for addition) (b) A + ( B + C ) = ( A + B ) + C. (Associate law for addition) (c) There is a unique m × n matrix O such that A + O = A (1) for any m × n matrix A . The matrix O is called the m × n additive identity or zero matrix. PROPERTIES OF MATRIX ADDITION 4 (d) For each m × n matrix A , there is a unique m × n matrix D such that A + D = O. (2) We shall write D as (- A ), so that (2) can be written as A + (- A ) = O. The matrix (- A ) is called the additive inverse or negative of A . Proof (a) To establish (a), we must prove that the i,j th element of A + B equals the i,j th element of B + A . The i,j th element of A + B is a ij + b ij ; the i,j th element of B + A . The i,j th element of B + A is b ij + a ij . Since the elements PROPERTIES OF MATRIX ADDITION 5 a ij are real (or complex) numbers, a ij + b ij = b ij + a ij (1 ≤ i ≤ m, 1 ≤ j ≤ n ) , the result follows. (b) Exercise. (c) Let U = [ u ij ] . Then A + U = A if and only if a ij + u ij = a ij , which holds if and only if u ij = 0 . Thus U is the m × n matrix all of whose entries are zero; U is denoted by O . (d) Exercise. PROPERTIES OF MATRIX ADDITION 6 Example 1 To illustrate (c) of Theorem 0.1, we note that the 2 × 2 zero matrix is 0 0 0 0 . If A = 4- 1 2 3 , we have 4- 1 2 3 + 0 0 0 0 = 4 + 0- 1 + 0 2 + 0 3 + 0 = 4- 1 2 3 . PROPERTIES OF MATRIX ADDITION 7 The 2 × 3 zero matrix is 0 0 0 0 0 0 . Example 2 To illustrate (d) of Theorem 0.1, let A = 2 3 4- 4 5- 2 . Then- A = - 2- 3- 4 4- 5 2 . We now have A + (- A ) = O . PROPERTIES OF MATRIX ADDITION 8 Example 3 Let A = 3- 2 5- 1 2 3 and B = 2 3 2- 3 4 6 ....
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Lecture_1_1 - Lecture Note 1: Sept 11 - 15, 2006 Dr. Jeff...

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