Lecture Notes

Lecture Notes - 1 NOTES FOR LINEAR ALGEBRA 133 William J....

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Unformatted text preview: 1 NOTES FOR LINEAR ALGEBRA 133 William J. Anderson McGill University These are not official notes for Math 133. They are intended for Andersons section 4, and are identical to the notes projected in class. 2 Contents 1 Linear Equations and Matrices. 5 1.1 Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Homogeneous Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Matrix Multiplication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Matrix Inverses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Elementary Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.6 Block Multiplication of Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Determinants and Eigenvalues. 19 2.1 Determinants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 Vector Geometry 29 3.1 Dot Product and Projections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Lines and Planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.1 Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 The Cross Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Matrix Transformations of R 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 4 CONTENTS Chapter 1 Linear Equations and Matrices. 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, 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....
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This note was uploaded on 12/01/2010 for the course MATH 133 taught by Professor Klemes during the Fall '08 term at McGill.

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Lecture Notes - 1 NOTES FOR LINEAR ALGEBRA 133 William J....

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