Lecture_0 - Lecture Note 0: Sept 5 - 8, 2006 Dr. Jeff...

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Unformatted text preview: Lecture Note 0: Sept 5 - 8, 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 Lecture Notes Go to http://www.math.cuhk.edu.hk/ ∼ jwong User Name : mat2310 Password : linear2006 Produced by Jeff Chak-Fu WONG 2 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 LINEAR EQUATIONS AND MATRICES 3 L INEAR S YSTEMS LINEAR SYSTEMS 4 An equation of the type a x = b expressing the variable b in terms of the variable x and the constant a , is called linear equation . The word linear is used here because the graph of the equation above is a straight line . Likewise, the equation a 1 x 1 + a 2 x 2 + ··· + a n x n = b, (1) expressing b in terms of the variables x 1 ,x 2 , ··· ,x n and the known constants a 1 ,a 2 , ··· ,a n and must find numbers x 1 ,x 2 , ··· ,x n , called unknowns , satisfying Eq. (1). LINEAR SYSTEMS 5 A solution to a linear equation Eq. (1) is a sequence of n numbers s 1 ,s 2 , ··· ,s n , which has the property that Eq. (1) is satisfied when x 1 = s 1 , x 2 = s 2 , ··· , x n = s n are substituted in Eq. (1). For example, x 1 = 2 , x 2 = 3 , and x 3 = 4 is a solution to the linear equation 6 x 1- 3 x 2 + 4 x 3 =- 13 , because 6(2)- 3(3) + 4(- 4) =- 13 . This is not the only solution to the given linear equation, since x 1 = 3 , x 2 = 1 , and x 3 =- 7 is another solution. LINEAR SYSTEMS 6 More generally, a system of m linear equations in n unknowns x 1 ,x 2 , ··· ,x n , or simply a linear system , is a set of m linear equations each in n unknowns. A linear system can be conveniently denoted by a 11 x 1 + a 12 x 2 + ··· + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ··· + a 2 n x n = b 2 . . . a ( m- 1)1 x 1 + a ( m- 1)2 x 2 + ··· + a ( m- 1) n x n = b m- 1 a m 1 x 1 + a m 2 x 2 + ··· + a mn x n = b m . (2) LINEAR SYSTEMS 7 The two subscripts i and j are used as follows. The first subscript i indicates that we are dealing with the i th equation, while the second subscript j is associated with the j th variable x j . Thus the i th equation is a i 1 x 1 + a i 2 x 2 + ··· + a in x n = b i . In Eq. (2) the a ij are known constants. Given values of b 1 ,b 2 , ··· ,b m , we want to find values of x 1 ,x 2 , ··· ,x n that will satisfy each equation in Eq. (2). A solution to a linear system Eq. (2) is a sequence of n numbers s 1 ,s 2 , ··· ,s n , which has the property that each equation in Eq. (2) is satisfied when x 1 = s 1 , s 2 = s 2 , ··· , x n = s n are substituted in Eq. (2)....
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Lecture_0 - Lecture Note 0: Sept 5 - 8, 2006 Dr. Jeff...

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