Notes for Linear Algebra 133

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

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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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4 CONTENTS Chapter 1 Linear Equations and Matrices....
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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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Notes for Linear Algebra 133 - 1 NOTES FOR LINEAR ALGEBRA...

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