Lecture on Algebra

Lecture on Algebra - Hanoi University of Technology Faculty...

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Hanoi University of Technology Faculty of Applied mathematics and informatics Advanced Training Program Lecture on Algebra Dr. Nguyen Thieu Huy Hanoi 2008
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Nguyen Thieu Huy, Lecture on Algebra Preface This Lecture on Algebra is written for students of Advanced Training Programs of Mechatronics (from California State University –CSU Chico) and Material Science (from University of Illinois- UIUC). When preparing the manuscript of this lecture, we have to combine the two syllabuses of two courses on Algebra of the two programs (Math 031 of CSU Chico and Math 225 of UIUC). There are some differences between the two syllabuses, e.g., there is no module of algebraic structures and complex numbers in Math 225, and no module of orthogonal projections and least square approximations in Math 031, etc. Therefore, for sake of completeness, this lecture provides all the modules of knowledge which are given in both syllabuses. Students will be introduced to the theory and applications of matrices and systems of linear equations, vector spaces, linear transformations, eigenvalue problems, Euclidean spaces, orthogonal projections and least square approximations, as they arise, for instance, from electrical networks, frameworks in mechanics, processes in statistics and linear models, systems of linear differential equations and so on. The lecture is organized in such a way that the students can comprehend the most useful knowledge of linear algebra and its applications to engineering problems. We would like to thank Prof. Tran Viet Dung for his careful reading of the manuscript. His comments and remarks lead to better appearance of this lecture. We also thank Dr. Nguyen Huu Tien, Dr. Tran Xuan Tiep and all the lecturers of Faculty of Applied Mathematics and Informatics for their inspiration and support during the preparation of the lecture. Hanoi, October 20, 2008 Dr. Nguyen Thieu Huy 1
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Nguyen Thieu Huy, Lecture on Algebra Contents Chapter 1: Sets. ................................................................................. 4 I. Concepts and basic operations. .......................................................................................... 4 II. Set equalities . ................................................................................................................... 7 III. Cartesian products. .......................................................................................................... 8 Chapter 2: Mappings. ....................................................................... 9 I. Definition and examples. ................................................................................................... 9 II. Compositions. ................................................................................................................... 9 III. Images and inverse images . .......................................................................................... 10 IV. Injective, surjective, bijective, and inverse mappings . ................................................. 11 Chapter 3: Algebraic Structures and Complex Numbers. .......... 13 I. Groups . ............................................................................................................................ 13 II. Rings. .............................................................................................................................. 15 III. Fields. ............................................................................................................................ 16 IV. The field of complex numbers. ..................................................................................... 16 Chapter 4: Matrices. ....................................................................... 26 I. Basic concepts . ................................................................................................................ 26 II. Matrix addition, scalar multiplication . ........................................................................... 28 III. Matrix multiplications. .................................................................................................. 29 IV. Special matrices. ........................................................................................................... 31 V. Systems of linear equations. ........................................................................................... 33 VI. Gauss elimination method . ........................................................................................... 34 Chapter 5: Vector spaces . .............................................................. 41 I. Basic concepts . ................................................................................................................ 41 II. Subspaces . ...................................................................................................................... 43 III. Linear combinations, linear spans. ................................................................................ 44 IV. Linear dependence and independence . ......................................................................... 45 V. Bases and dimension. ..................................................................................................... 47 VI. Rank of matrices. .......................................................................................................... 50 VII. Fundamental theorem of systems of linear equations . ................................................ 53 VIII. Inverse of a matrix . .................................................................................................... 55 X. Determinant and inverse of a matrix, Cramer’s rule. ..................................................... 60 XI. Coordinates in vector spaces . ....................................................................................... 62 Chapter 6: Linear Mappings and Transformations. ................... 65 I. Basic definitions . ............................................................................................................. 65 II. Kernels and images . ....................................................................................................... 67 III. Matrices and linear mappings . ...................................................................................... 71 IV. Eigenvalues and eigenvectors. ...................................................................................... 74 V. Diagonalizations.
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This note was uploaded on 11/07/2009 for the course FOFL 4531 taught by Professor Haiza during the Spring '09 term at Carlos Albizu.

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Lecture on Algebra - Hanoi University of Technology Faculty...

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