{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Elementary Linear Algebra( K.R. Matthews)

Elementary Linear Algebra( K.R. Matthews) - ELEMENTARY...

Info icon This preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Second Online Version, December 1998 Comments to the author at [email protected]
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Contents 1 LINEAR EQUATIONS 1 1.1 Introduction to linear equations . . . . . . . . . . . . . . . . . 1 1.2 Solving linear equations . . . . . . . . . . . . . . . . . . . . . 6 1.3 The Gauss–Jordan algorithm . . . . . . . . . . . . . . . . . . 8 1.4 Systematic solution of linear systems. . . . . . . . . . . . . . 9 1.5 Homogeneous systems . . . . . . . . . . . . . . . . . . . . . . 16 1.6 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 MATRICES 23 2.1 Matrix arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Linear transformations . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . 31 2.4 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5 Non–singular matrices . . . . . . . . . . . . . . . . . . . . . . 36 2.6 Least squares solution of equations . . . . . . . . . . . . . . . 47 2.7 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3 SUBSPACES 55 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Subspaces of F n . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Linear dependence . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4 Basis of a subspace . . . . . . . . . . . . . . . . . . . . . . . . 61 3.5 Rank and nullity of a matrix . . . . . . . . . . . . . . . . . . 64 3.6 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4 DETERMINANTS 71 4.1 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 i
Image of page 2
5 COMPLEX NUMBERS 89 5.1 Constructing the complex numbers . . . . . . . . . . . . . . . 89 5.2 Calculating with complex numbers . . . . . . . . . . . . . . . 91 5.3 Geometric representation of C . . . . . . . . . . . . . . . . . . 95 5.4 Complex conjugate . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5 Modulus of a complex number . . . . . . . . . . . . . . . . . 99 5.6 Argument of a complex number . . . . . . . . . . . . . . . . . 103 5.7 De Moivre’s theorem . . . . . . . . . . . . . . . . . . . . . . . 107 5.8 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6 EIGENVALUES AND EIGENVECTORS 115 6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Definitions and examples . . . . . . . . . . . . . . . . . . . . . 118 6.3 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7 Identifying second degree equations 129 7.1 The eigenvalue method . . . . . . . . . . . . . . . . . . . . . . 129 7.2 A classification algorithm . . . . . . . . . . . . . . . . . . . . 141 7.3 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8 THREE–DIMENSIONAL GEOMETRY 149 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.2 Three–dimensional space . . . . . . . . . . . . . . . . . . . . . 154 8.3 Dot product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.4 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8.5 The angle between two vectors . . . . . . . . . . . . . . . . . 166 8.6 The cross–product of two vectors . . . . . . . . . . . . . . . . 172 8.7 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 8.8 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 9 FURTHER READING 189 ii
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
List of Figures 1.1 Gauss–Jordan algorithm . . . . . . . . . . . . . . . . . . . . . 10 2.1 Reflection in a line . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Projection on a line . . . . . . . . . . . . . . . . . . . . . . . 30 4.1 Area of triangle OPQ . . . . . . . . . . . . . . . . . . . . . . . 72 5.1 Complex addition and subtraction . . . . . . . . . . . . . . . 96 5.2 Complex conjugate . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3 Modulus of a complex number . . . . . . . . . . . . . . . . . 99 5.4 Apollonius circles . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.5 Argument of a complex number . . . . . . . . . . . . . . . . . 104 5.6 Argument examples . . . . . . . . . . . . . . . . . . . . . . . 105 5.7 The n th roots of unity . . . . . . . . . . . . . . . . . . . . . . . 108 5.8 The roots of z n = a . . . . . . . . . . . . . . . . . . . . . . . . 109 6.1 Rotating the axes . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.1 An ellipse example . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2 ellipse: standard form . . . . . . . . . . . . . . . . . . . . . . 137 7.3 hyperbola: standard forms . . . . . . . . . . . . . . . . . . . . 138 7.4 parabola: standard forms (i) and (ii) . . . . . . . . . . . . . . 138 7.5 parabola: standard forms (iii) and (iv) . . . . . . . . . . . . . 139 7.6 1st parabola example . . . . . . . . . . . . . . . . . . . . . . . 140 7.7 2nd parabola example . . . . . . . . . . . . . . . . . . . . . . 141 8.1 Equality and addition of vectors . . . . . . . . . . . . . . . . 150 8.2 Scalar multiplication of vectors . . . . . . . . . . . . . . . . . . 151 8.3 Representation of three–dimensional space . . . . . . . . . . . 155 8.4 The vector AB . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.5 The negative of a vector . . . . . . . . . . . . . . . . . . . . . . 157 iii
Image of page 4
1 8.6 (a) Equality of vectors; (b) Addition and subtraction of vectors.157 8.7 Position vector as a linear combination of i , j and k . . . . . . 158 8.8 Representation of a line . . . . . . . . . . . . . . . . . . . . . . 162 8.9 The line AB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 8.10 The cosine rule for a triangle . . . . . . . . . . . . . . . . . . . 167 8.11 Pythagoras’ theorem for a right–angled triangle. . . . . . . . 168 8.12 Distance from a point to a line . . . . . . . . . . . . . . . . . . 169 8.13 Projecting a segment onto a line . . . . . . . . . . . . . . . . . 171 8.14 The vector cross–product . . . . . . . . . . . . . . . . . . . . . 174 8.15 Vector equation for the plane ABC . . . . . . . . . . . . . . . 177 8.16 Normal equation of the plane ABC . . . . . . . . . . . . . . . 178 8.17 The plane ax + by + cz = d . . . . . . . . . . . . . . . . . . . . 179 8.18 Line of intersection of two planes . . . . . . . . . . . . . . . . . 182 8.19 Distance from a point to the plane ax + by + cz = d . . . . . . 184
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2
Image of page 6
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1 , x 2 , · · · , x n is an equation of the form a 1 x 1 + a 2 x 2 + · · · + a n x n = b, where a 1 , a 2 , . . . , a n , b are given real numbers.
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern