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mat 2310 0607_2_note0_1

mat 2310 0607_2_note0_1 - Lecture Note 0 Jan 8 2007 Dr Jeff...

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Lecture Note 0: Jan 8, 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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Lecture Notes Go to http://www.math.cuhk.edu.hk/ jwong User Name : mat2310c Password : 2007 Produced by Jeff Chak-Fu WONG 2
Tutorial Class Tuesday : MMW: 707 Wednesday : MMW: 706 Produced by Jeff Chak-Fu WONG 3

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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 L INEAR E QUATIONS AND M ATRICES 4
L INEAR S YSTEMS L INEAR S YSTEMS 5

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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). L INEAR S YSTEMS 6
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. L INEAR S YSTEMS 7

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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) L INEAR S YSTEMS 8
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). L INEAR S YSTEMS 9

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To find solutions to a linear system, we shall use a technique called the method of elimination . That is, we eliminate some of the unknowns by adding a multiple of one equation to another equation.
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