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

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Lecture Note 1: Sept 11 - 15, 2006 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2006 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. LU-Factorization L INEAR E QUATIONS AND M ATRICES 2
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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 P ROPERTIES OF M ATRIX O PERATIONS 3
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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. P ROPERTIES OF M ATRIX A DDITION 4
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(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 P ROPERTIES OF M ATRIX A DDITION 5
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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. P ROPERTIES OF M ATRIX A DDITION 6
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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 . P ROPERTIES OF M ATRIX A DDITION 7
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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 . P ROPERTIES OF M ATRIX A DDITION 8
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Example 3 Let A = 3 - 2 5 - 1 2 3 and B = 2 3 2 - 3 4 6 . Then A - B = 3 - 2 - 2 - 3 5 - 2 - 1 + 3 2 - 4 3 - 6 = 1 - 5 3 2 - 2 - 3 . P ROPERTIES OF M ATRIX A DDITION 9
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P ROPERTIES OF M ATRIX M ULTIPLICATION Theorem 0.2 (Properties of Matrix Multiplication) (a) If A, B and C are of the appropriate sizes, then A ( BC ) = ( AB ) C.
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