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Unformatted text preview: 1 NOTES FOR MATH 123 LINEAR ALGEBRA and PROBABILITY William J. Anderson McGill University These are not official notes for Math 123. They are intended for Andersons section, and are identical to the notes projected in class. 2 Contents 1 Systems of Linear Equations and Matrices. 5 1.1 Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Matrix Multiplication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4 Matrix Inverses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5 Elementary Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.6 Block Multiplication of Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 4 CONTENTS Chapter 1 Systems of Linear Equations and Matrices. Reference. Finite Mathematics with Calculus and Applications , 8th Ed., Chapter 2. 1.1 Linear Equations. Introduction. x + 2 y =  4 is a linear equation. A solution is a pair s 1 , s 2 of numbers such that s 1 + 2 s 2 =  4 . s 1 = 1 , s 2 =  5 / 2 is a solution. So is s 1 =  1 , s 2 =  3 / 2 . In fact, there are infinitely many solutions given by s 1 = s , s 2 =  s + 4 2 for any number s . This is called a parametrized solution. The pair x + 2 y =  4 2 x + 3 y = 14 (1.1) is called a system of linear equations. Replacing the 2nd equation by the second minus twice the first gives x + 2 y =  4 y = 22 so we conclude that x = 40 , y =  22 . We just did Gaussian elimination (also called the echelon method), and the last display is the system in echelon form (except for the 1 ). In this case the system has a unique solution. But if the system was x + 2 y =  4 2 x + 4 y = 14 , there would be no solution. And if the system were x + 2 y =  4 2 x + 4 y =  8 , there would be infinitely many solutions as in the example at the very beginning. 5 6 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES. Linear Systems. An equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1 , . . . , a n , and b are given numbers is called a linear equation in the unknown variables x 1 , . . . , x n . A set of numbers s 1 , . . . , s n is called a solution to this equation if a 1 s 1 + a 2 s 2 + + a n s n = b. Equivalently, we arrange the s i s into a column vector X = s 1 s 2 . . . s n , and call X a solution. m of these linear equations 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 ....
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 Spring '09
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