Math 210 Linear Algebra Fall 2014
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1 Math 210 Linear Algebra Fall 2014 Instructor's Name: Office Location: Office Hours: Office Phone: Course Description This is a first course in vectors, matrices, vector spaces, and linear transformations. The ideas in this course serve as an introduction to more abstract mathematics courses at the junior-senior level, and also covers many useful applications outside mathematics. Topics include: vectors, operations on matrices, inverse of a matrix, solution of systems of linear equations, rank of matrix, vector spaces and subspaces, linear dependence and independence, basis and dimension, linear transformation, sums, composites, inverses of linear transformations, range and kernel of a linear transformation, determinants, eigenvalues and eigenvectors, orthogonality and inner product spaces, and real quadratic forms. Illinois Articulation Initiative (IAI) number: N/A Credit and Contact Hours: Lecture 3 Lab 0 Credit Hours 3 Prerequisites: Grade of "C" or better in MATH 172 or equivalent. Concurrent: MATH 172, MATH 220. Books, Supplies, and Supplementary Materials A. Textbooks Required: Elementary Linear Algebra, 11 1h Ed., 2013; Anton Student Solutions Manual (optional) Publisher: Wiley Required parameters are missing or incorrect.math 210 Course Syllabus Page 1
2 B. Other Required Materials None Methods of Instruction: Lecture Student Learning Outcomes: General Education Student Learning Outcomes: Objectives 1. Define "set" and the related terminology 2. Define "function" and the related terminology 3. Explain what is meant by and be able to form the composition of functions 4. Explain what is meant by a system of equations, a solution of the system, a consistent system, an inconsistent system and a homogeneous system of equations 5. Define "matrix. 6. Explain what is meant by an "m x n" matrix, a "square matrix of order n," the "(i,j) entry" of a matrix and the "main diagonal" of a square matrix 7. Define and determine the equality of two matrices, the sum of two matrices, the difference of two matrices, the product of two matrices, and the product of the scalar and a matrix 8. Use summation notation in the definition of matrix multiplication and proof of certain matrix properties 9. Define "transpose of a matrix" and find the transpose of a given matrix 10. Make a formal or informal proof of various theorems concerning the above objects and operations 11. State all of the algebraic properties of matrix operations as discussed in class 12. Prove selected algebraic properties of matrix operations as well as various theorems which are off shoots of these properties 13. State what is meant by the zero-matrix, by a diagonal matrix, a scalar matrix, and the identity matrix of order n 14. Define "upper triangular form" and "lower triangular form" for a matrix 15. Define "singular" matrix, "nonsingular" matrix, and "inverse" of a matrix and find the inverse of a given matrix when it arrives 16. Prove various theorems concerning the objects mentioned in Objectives Explain the connection between singular and nonsingular matrices to the solution of a system of equations 18. Define "n by n" elementary matrices of type I, II, or III 19. Prove selected theorems concerning the operation of elementary matrices on a given matrix 20. Use elementary matrices to develop a technique for finding the inverse of a given matrix Required parameters are missing or incorrect.math 210 Course Syllabus Page 2
3 21. Explain what is meant by row-reduced echelon form for a matrix and transform a given matrix into row-reduced echelon form 22. Define the three elementary row operations on a matrix 23. Explain what is meant by one matrix being row equivalent to a second matrix 24. Prove various theorems concerning row equivalence and row-reduced echelon form 25. Use matrix techniques discussed in class to solve systems of linear equations 26. Define "real vector space" and explain the significance of each of the components of the definition 27. Give examples of a vector space 28. Define "subspace of a vector space" and give examples. 29. Determine whether or not a given object is a vector space or subspace 30. Use appropriate notation, work problems, and prove selected theorems involving vector spaces and subspaces 31. Define "linear combination" of a set of vectors 32. State what is meant by a set of vectors "spanning" a vector space 33. Explain what is meant by a linearly dependent or linearly independent set of vectors 34. Define a "basis" for a vector space 35. Explain what is meant by a nonzero vector space 36. Define the dimension of a nonzero vector space 37. Give examples, use appropriate notation, work problems, and prove selected theorems concerning linear dependence and independence, bases, and dimensions of vector spaces 38. Define row space and column space of an m by n matrix 39. Explain what is meant by the row (column) rank of a matrix 40. Discuss the structure of a linear system of equations 41. Define the determinant of an n-by-n matrix and evaluate the determinant of a given matrix 42. Discuss and prove the various properties of determinants and use these properties to aid in solving problems involving determinants 43. Define the "minor" of an element a ij of a matrix A 44. Define the "cofactor" of an element a ij of a matrix A 45. Explain and perform the process of finding a determinant by cofactor expansion 46. Define the adjoint of a matrix A and find the adjoint of a given matrix 47. Use appropriate notation and prove selected theorems that demonstrate the connection among the inverse of a matrix, the determinant of a matrix, and the adjoint of a matrix 48. Apply determinants in other selected situations as discussed in class 49. Define the dot product of two vectors and discuss and/or prove its properties 50. State the Cauch-Schwarz inequality and the triangle inequnt by cofactor expansion 51. Define the distance between two vectors and what are orthogonal vectors 52. Explain what is meant by an orthogonal set of vectors and an orthonormal set of vectors Required parameters are missing or incorrect.math 210 Course Syllabus Page 3
4 53. Define and calculate the scalar projection and vector projection of one vector on another 54. Use appropriate notation, work problems, and prove selected theorems concerning inner products, the Cauch-Schwarz and triangle inequalities, distance and orthogonality 55. Discuss and use the Gram-Schmidt Process for vectors 56. Define "linear transformation" of a vector space V into a vector space W 57. State what is meant by the null space and range of a linear transformation 58. Explain what is meant by the matrix representation of a linear transformation 59. Find the matrix representation of a given linear transformation 60. Define the "sum," "scalar multiple" and "composition" of linear transformations and thereby define a vector space of linear transformations 61. State what is meant by a vector space of matrices 62. Explain the concept of a coordinate vector with respect to an ordered basis 63. Find how coordinate vectors transform under a change of basis 64. Define "similar matrices 65. Give examples, use appropriate notation, work problems, and prove selected theorems concerning rank of a matrix, linear transformations, null spaces, ranges, vector spaces of linear transformations, and vector spaces of matrices 66. Define "diagonalizable linear transformation" and give example space of matrices 67. Define "eigenvalue" and "eigenvector" of a linear transformation, give examples, and find the eigenvalues of eigenvectors of a given matrix 68. State what is meant by the characteristic polynomial of a matrix 69. Work problems based on the definitions mentioned in objectives and theorems based on those definitions 70. Explain what is meant by a symmetric matrix and skew symmetric matrix, and determine whether or not a given matrix is symmetric or skew symmetric 71. Discuss and/or prove the theorems connecting diagonalization and symmetric matrices 72. Define "Real Quadratic Form" and "equivalence" of real quadratic forms 73. Explain what is meant by congruent matrices 74. Use appropriate notation, work problems and prove selected theorems involving quadratic forms TOPICAL OUTLINE Week Topic or Class Activity 1 Sets, Functions, Matrices 2 More matrices, solving systems of equations 3 Vector spaces Required parameters are missing or incorrect.math 210 Course Syllabus Page 4
5 4 More on vector spaces, linear independence 5 Spanning sets, bases, and finite dimensional vector spaces 6 Rank of a matrix, structure of solutions of a system of equations 7 Determinants 8 More on determinants, dot products 9 Orthogonality, The Gram-Schmidt process 10 Linear Transformations and matrix representations 11 Operations on linear transformations, null space and range 12 Change of basis, more on matrix representation 13 Similar matrices, eignevalues and eigenvectors 14 Diagonalization and symmetric matrices 15 Applications Graded Assignments and Policies Graded Assignments Grading Policy The individual instructor will determine which items he or she considers essential for the student to memorize without error and test accordingly. Each instructor will set minimum standards for performance on tests. Major Tests and Quizzes The individual instructor will determine which items he or she considers essential for the student to memorize without error and test accordingly. Each instructor will set minimum standards for performance on tests. A comprehensive final examination will be given. Classroom Policies and Procedures Required parameters are missing or incorrect.math 210 Course Syllabus Page 5
6 General Information Attendance Policy Make-up Policy Extra-credit Policy Final Exam Information A comprehensive final examination will be given. Academic Honor Code The objective of the academic honor code is to sustain a learning-centered environment in which all students are expected to demonstrate integrity, honor, and responsibility, and recognize the importance of being accountable for one s academic behavior. College Statement about grades of F and Withdrawal from Class Students may withdraw from a course by processing an add/drop form during regular office hours through the Registration and Records Office at Main Campus or Romeoville Campus, or by phone at Please note the withdrawal dates listed on your bill or student schedule. Every course has its own withdrawal date. Failure to withdraw properly may result in a failing grade of F in the course. At any time prior to the deadline dates established, an instructor may withdraw a student from class because of poor attendance, poor academic performance or inappropriate academic behavior, such as, but not limited to, cheating or plagiarism. Intellectual Property Students own and hold the copyright to the original work they produce in class. It is a widely accepted practice to use student work as part of the college s internal selfevaluation, assessment procedures, or other efforts to improve teaching and learning and in promoting programs and recruiting new students. If you do not wish your work to be used in this manner, please inform the instructor. Student Code of Conduct Each student is responsible for reading and adhering to the Student Code of Conduct as stated in the college catalog. Sexual Harassment Joliet Junior College seeks to foster a community environment in Required parameters are missing or incorrect.math 210 Course Syllabus Page 6
7 which all members respect and trust each other. In a community in which persons respect and trust each other, there is no place for sexual harassment. JJC has a strong policy prohibiting the sexual harassment of one member of the college community by another. See the Catalog or Student Handbook. Student Support Prepared by: Reviewed by: Prof. Robert Tuskey Prof. Jean McArthur Department Chair Date Required parameters are missing or incorrect.math 210 Course Syllabus Page 7
8 Revised 04/11 Reviewed 08/09 Revised 11/02 Revised 11/98 Revised 10/93 Revised 08/96 Revised 02/92 Revised 11/91 Revised 11/89 Required parameters are missing or incorrect.math 210 Course Syllabus Page 8
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