LECTURES ON APPLIED MATHEMATICS

LECTURES ON APPLIED MATHEMATICS Part 1: Linear Algebra Ray M. Bowen Former Professor of Mechanical Engineering President Emeritus Texas A&M University College Station, Texas Copyright Ray M. Bowen Updated February, 2013 ____________________________________________________________________________ PREFACE To Part 1 It is common for Departments of Mathematics to offer a junior-senior level course on Linear Algebra. This book represents one possible course. It evolved from my teaching a junior level course at Texas A&M University during the several years I taught after I served as President. I am deeply grateful to the A&M Department of Mathematics for allowing this Mechanical Engineer to teach their students. This book is influenced by my earlier textbook with C.-C Wang, Introductions to Vectors and Tensors, Linear and Multilinear Algebra. This book is more elementary and is more applied than the earlier book. However, my impression is that this book presents linear algebra in a form that is somewhat more advanced than one finds in contemporary undergraduate linear algebra courses. In any case, my classroom experience with this book is that it was well received by most students. As usual with the development of a textbook, the students that endured its evolution are due a statement of gratitude for their help. As has been my practice with earlier books, this book is available for free download at the site http://www1.mengr.tamu.edu/rbowen/ or, equivalently, from the Texas A&M University Digital Library’s faculty repository, http://repository.tamu.edu/handle/1969.1/2500. It is inevitable that the book will contain a variety of errors, typographical and otherwise. Emails to rbowen@tamu.edu that identify errors will always be welcome. For as long as mind and body will allow, this information will allow me to make corrections and post updated versions of the book. College Station, Texas R.M.B. Posted January, 2013 ii ______________________________________________________________________________ CONTENTS Part 1 Linear Algebra Selected Readings for Part I………………………………………………………… 2 CHAPTER 1 Elementary Matrix Theory……………………………………… 3 Section 1.1 Section 1.2 Section 1.3 Section 1.4 Section 1.5 Section 1.6 Section 1.7 Section 1.8 Section 1.9 Section 1.10 Section 1.11 Basic Matrix Operations………………………………………... 3 Systems of Linear Equations…………………………………… 13 Systems of Linear Equations: Gaussian Elimination…………... 21 Elementary Row Operations, Elementary Matrices…………….. 39 Gauss-Jordan Elimination, Reduced Row Echelon Form……… 45 Elementary Matrices-More Properties…………………………. 53 LU Decomposition……………………………………………. 69 Consistency Theorem for Linear Systems……………………… 91 The Transpose of a Matrix……………………………………… 95 The Determinant of a Square Matrix…………………………….101 Systems of Linear Equations: Cramer’s Rule ……………… …125 CHAPTER 2 Vector Spaces…………………………………………… 131 Section 2.1 Section 2.2 Section 2.3 Section 2.4 Section 2.5 Section 2.6 Section 2.7 The Axioms for a Vector Space…………………………….. 131 Some Properties of a Vector Space……………………….. 139 Subspace of a Vector Space…………………………….. 143 Linear Independence……………………………………. 147 Basis and Dimension…………………………………… 163 Change of Basis………………………………………… 169 Image Space, Rank and Kernel of a Matrix………………… 181 CHAPTER 3 Linear Transformations…………………………………….. 207 Section 3.1 Section 3.2 Section 3.3 Section 3.4 Section 3.5 Section 3.6 Definition of a Linear Transformation…………….. 207 Matrix Representation of a Linear Transformation.. 211 Properties of a Linear Transformation……………. 217 Sums and Products of Linear Transformations….. 225 One to One Onto Linear Transformations………. 231 Change of Basis for Linear Transformations........... 235 CHAPTER 4 Vector Spaces with Inner Product………………………... 247 Section 4.1 Definition of an Inner Product Space…………….. 247 iii Section 4.2 Section 4.3 Section 4.4 Section 4.5 Section 4.6 Section 4.7 Section 4.8 Section 4.9 Section 4.10 Section 4.11 Section 4.12 Section 4.13 Section 4.14 Section 4.14 Schwarz Inequality and Triangle Inequality……… 255 Orthogonal Vectors and Orthonormal Bases…….. 263 Orthonormal Bases in Three Dimensions……….. 277 Euler Angles……………………………………... 289 Cross Products on Three Dimensional Inner Product Spaces 295 Reciprocal Bases………………………………… 301 Reciprocal Bases and Linear Transformations…. 311 The Adjoint Linear Transformation……………. 317 Norm of a Linear Transformation……………… 329 More About Linear Transformations on Inner Product Spaces 333 Fundamental Subspaces Theorem……………... 343 Least Squares Problem………………………… 351 Least Squares Problems and Overdetermined Systems 357 A Curve Fit Example…………………………… 373 CHAPTER 5 Eigenvalue Problems…………………………………… 387 Section 5.1 Section 5.2 Section 5.3 Section 5.4 Section 5.5 Section 5.6 Section 5.7 Eigenvalue Problem Definition and Examples… 387 The Characteristic Polynomial…………………. 395 Numerical Examples…………………………… 403 Some General Theorems for the Eigenvalue Problem 421 Constant Coefficient Linear Ordinary Differential Equations 431 General Solution……………………………….. 435 Particular Solution……………………………… 453 CHAPTER 6 Additional Topics Relating to Eigenvalue Problems…… 467 Section 6.1 Section 6.2 Section 6.3 Section 6.4 Section 6.5 Section 6.6 Section 6.7 Section 6.8 Section 6.9 Characteristic Polynomial and Fundamental Invariants 467 The Cayley-Hamilton Theorem………………… 471 The Exponential Linear Transformation……….. 479 More About the Exponential Linear Transformation 493 Application of the Exponential Linear Transformation 499 Projections and Spectral Decompositions………. 511 Tensor Product of Vectors……………………….. 525 Singular Value Decompositions………………… 531 The Polar Decomposition Theorem…………….. 555 INDEX………………………………………………………………….. vii iv PART I1 NUMERICAL ANALYSIS Selected Readings for Part II………………………………………… PART III. ORDINARY DIFFERENTIAL EQUATIONS Selected Readings for Part III………………………………………… PART IV. PARTIAL DIFFERENTIAL EQUATIONS Selected Readings for Part IV………………………………………… v ... - tailieumienphi.vn