Notes for Linear Algebra 133

Notes for Linear Algebra 133 - 1 NOTES FOR LINEAR ALGEBRA...

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1 NOTES FOR LINEAR ALGEBRA 133 William J. Anderson McGill University These are not official notes for Math 133. They are intended for Anderson’s section 4, and are identical to the notes projected in class.
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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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Elementary Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.6 Block Multiplication of Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 Determinants and Eigenvalues. 21 2.1 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Vector Geometry 31 3.1 Dot Product and Projections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.1 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 The Cross Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 Matrix Transformations of R 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4.1 Composite of Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.4.2 Inverse of a Matrix Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4 The Vector Space R n . 47 4.1 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.1.1 Invertibility of Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1.2 Linear Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2.1 Existence of Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.1 The Rank Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.2 Null Space and Image Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3
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4 CONTENTS
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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. 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. 5
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6 CHAPTER 1. LINEAR EQUATIONS AND MATRICES. 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 . . . a m 1 x 1 + a m 2 x 2 + · · · + a mn x n = b m is called a system of m equations in n unknowns. The column vector X is a solution if X is a solution to each of the equations. It is possible that a system of linear equations (1) has no solution (then the system is called inconsistent), (2) has a unique solution, (3) has infinitely many solutions. The system is called consistent if there is at least one solution. Otherwise inconsistent .
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