Columbus State Community College Mathematics Department Course and Number: MATH 2174 - Linear Algebra and Differential Equations for Engineering CREDITS: 5 CLASS HOURS PER WEEK: 5 PREREQUISITES: MATH 2173 with a C or higher DESCRIPTION OF COURSE (AS IT APPEARS IN THE COLLEGE CATALOG): Matrix theory, eigenvectors and eigenvalues, ordinary and partial differential equations. COURSE GOALS AND/OR OBJECTIVES To introduce to the student the concepts, methods, and applications of topics in linear algebra and differential equations necessary for further study in engineering; to present key ideas and concepts from a variety of perspectives; to develop student s mathematical thinking and problem solving ability. LEARNING OUTCOMES: 1. Represent systems of linear equations in matrix form and solve them. 2. Perform elementary row operations to reduce a matrix to (reduced) echelon form. 3. Understand algebraic and geometric representations of vectors in R n and their operations. Identify independent and dependent sets of vectors. 4. Compute the product of a matrix and a vector. Employ matrix reduction techniques to both solve systems of linear equations and identify inconsistent and dependent systems. 5. Understand and use linear transformations: find the image of a given vector under a given linear transformation, express a given linear transformation as a matrix & find the standard matrix of a given linear transformation, determine if a linear transformation is one-to-one, and apply concepts to problems in engineering. 6. Understand and use matrix algebra: compute the transpose and inverse of a matrix, the product of a scalar and a matrix, and sums & differences of matrices; multiply matrices, understand associativity and noncommutativity of matrix multiplication, and interpret a matrix product as a composition of linear transformations; solve systems of linear equations using the inverse of the coefficient matrix; understand the invertible matrix theorem. 7. Compute the determinant of a n nmatrix using cofactor expansion (and row reduction when appropriate) and understand various properties of determinants including the determinant of product & how to use determinants to decide if a matrix is invertible. 8. Determine if a given number is an eigenvalue of a given matrix and if so, find its corresponding eigenvector. Determine if a given vector is an eigenvector of a given matrix and if so, find its corresponding eigenvalue. 9. Find the characteristic polynomial and the eigenvalues of a given matrix. 10. Understand the concept of a diagonalizable matrix and use eigenvectors to diagonalize a matrix. 11. Represent a linear transformation by a matrix relative to given bases. 12. Determine the complex eigenvalues of a given matrix, and a basis for the corresponding eigenspace.
13. Determine the long-term behavior of various dynamical systems. 14. Understand the basic theory of systems of first order linear equations. Find direction fields for and solve systems of ODEs, solve IVPs, and be able to express general solutions in terms of real-valued functions when the coefficient matrix has complex eigenvalues. 15. Construct the fundamental matrix for a given system of equations. 16. Find a fundamental set of solutions to systems that have a coefficient matrix with repeated eigenvalues. 17. Solve two-point boundary value problems or show that no solution exists. Find the eigenvalues and eigenfunctions of given boundary value problems. 18. Determine if a given function is periodic and, if so, find its fundamental period. Find Fourier series and understand the Fourier Convergence Theorem. Understand the definitions and basic properties of even and odd functions and find Fourier cosine and sine series. 19. Use the method of separation of variables to solve the heat equation for one space variable and solve heat conduction problems with various boundary conditions. 20. Solve the wave equation and problems involving vibrations of an elastic string. GENERAL EDUCATION GOALS: Critical thinking and Quantitative Literacy EQUIPMENT AND MATERIAL REQUIRED: A Graphing Calculator is recommended. However, no symbolic manipulator, e.g. TI-89, TI-92 etc., is permitted. TEXTBOOK, MANUALS, REFERENCES, AND OTHER READINGS: Linear Algebra and its Applications, Fourth Edition, David C. Lay, Pearson Addison- Wesley, 2012 Elementary Differential Equations and Boundary Value Problems, Ninth edition, William E. Boyce and Richard C. DiPrima, Wiley and Sons, Inc., 2009 GENERAL INSTRUCTIONAL METHODS: Classroom lecture, discussion, recitation, and/or problem solving explorations supplemented by visual and/or computer aids. STANDARDS AND METHODS FOR EVALUATION: Grades will NOT be curved, skewed, or otherwise inflated and no retests are to be given. Three tests are recommended. The final examination should weigh between 25% and 35% of the course grade; preferably about 30%. GRADING SCALE: Letter grades for the course will be awarded using a 90% - 80% - 70% - 60% scale. SPECIAL COURSE REQUIREMENTS: None UNITS OF INSTRUCTION
Please provide a weekly course schedule indicating the Units of Instruction, learning objectives/goals, assigned readings, assignments, and exams. Unit 1 - Unit of Instruction: Linear Equations in Linear Algebra 1. Represent systems of linear equations in matrix form. 2. Solve systems of linear equations. 3. Demonstrate the ability to perform elementary row operations to reduce a matrix to (reduced) echelon form. 4. Understand algebraic and geometric representations of vectors in R n and their operations, including addition and scalar multiplication. 5. Determine if a given vector is a linear combination of other given vectors. 6. Give a geometric interpretation for the span of a set of vectors. 7. Compute the product of a matrix and a vector. 8. Employ matrix reduction techniques to solve systems of linear equations and identify inconsistent and dependent systems. 9. Identify independent and dependent sets of vectors. 10. Find the image of a given vector under a given transformation. 11. Express a given linear transformation as a matrix. 12. Find the standard matrix of a given linear transformation. 13. Determine if a linear transformation is one-to-one. 14. Solve engineering applications. - Assigned Reading: 1.1-1.5, 1.7-1.10 (Lay) Unit 2 - Unit of Instruction: Matrix Algebra 1. Compute sums, scalar products, and differences using matrices. 2. Multiply matrices, and understand associativity and noncommutativity of matrix multiplication. 3. Interpret a matrix product as a composition of linear transformations. 4. Compute the transpose of a matrix. 5. Compute the inverse of an invertible matrix. 6. Solve systems of linear equations using the inverse of the coefficient matrix. 7. Understand the invertible matrix theorem. - Assigned Reading: 2.1-2.3 (Lay) Unit 3 - Unit of Instruction: Determinants
1. Compute the determinant of a square matrix using a cofactor expansion. 2. Compute the determinant of a square matrix using both row reduction and cofactor expansion. 3. Compute the determinant of a triangular matrix. 4. Understand and apply various properties of the determinant. 5. Use a determinant to determine if a matrix is invertible. - Assigned Reading: 3.1, 3.2 (Lay) Unit 4 - Unit of Instruction: Eigenvalues and Eigenvectors - Student Learning Outcomes: 1. Determine if a given number is an eigenvalue of a given matrix. If so, find its corresponding eigenvector. 2. Determine if a given vector is an eigenvector of a given matrix. If so, find its corresponding eigenvalue. 3. Find the characteristic polynomial and the eigenvalues of a given matrix. 4. Understand the concept of a diagonalizable matrix. 5. Use eigenvectors to diagonalize a matrix. 6. Represent a linear transformation by a matrix relative to given bases. 7. Determine the complex eigenvalues of a given matrix, and a basis for the corresponding eigenspace. 8. Determine the long-term behavior of various dynamical systems. - Assigned Reading: 5.1-5.6 (Lay) Unit 5 - Unit of Instruction: Systems of Differential Equations 1. Determine if a set of solutions of a system form a fundamental set of solutions on an interval, and if so, use them to construct the general solution to the system on that interval. 2. Draw directions fields for systems of ODEs. 3. Solve homogeneous linear systems with constant coefficients. 4. Construct real-valued solutions to homogeneous linear systems with constant coefficients when the coefficient matrix has complex eigenvalues. 5. Construct the fundamental matrix for a given system of equations. 6. Find a fundamental set of solutions to systems that have a coefficient matrix with repeated eigenvalues. - Assigned Reading: Sections 7.4-7.8 (Boyce/Diprima) - Assessment Methods: Final exam, tests, quizzes, graded HW, individual or group projects, etc.
Unit 6 - Unit of Instruction: Partial Differential Equations and Fourier Series 1. Solve two-point boundary value problems or show that no solution exists. 2. Find the eigenvalues and eigenfunctions of given boundary value problems. 3. Determine if a given function is periodic. If it is, find its fundamental period. 4. Find the Fourier series for a given function. 5. Describe how a Fourier series seems to be converging. 6. Find the Fourier series for a given function periodically extended outside a given interval. 7. Determine whether a given function is even, odd, or neither. 8. Given a function on an interval of length L, sketch the graphs of its even and odd extensions of period 2L. 9. Find Fourier Sine and Cosine Series. 10. Use the method of separation of variables to solve the heat equation for one space variable. 11. Solve heat conduction problems with various boundary conditions. 12. Solve the wave equation and problems involving vibrations of an elastic string. - Assigned Reading: Sections 10.1-10.7 (Boyce/Diprima) - Assessment Methods: Final exam, tests, quizzes, graded HW, individual or group projects, etc.
MATH 2174 - Linear Algebra and Differential Equations for Engineering (Two day per week calendar) Week Day Sections Title Comments 1 1.1 Systems of Linear Equations 1 1.2 Row Reduction and Echelon Forms 2 1.3 Vector Equations 2 3 1.4 The Matrix Equation Ax=b Don t introduce matrix multiplication until Sec. 2.1. 4 1.5 Solution Sets of Linear Systems 1.7 Linear Independence 5 1.8 Introduction to Linear Transformations 3 6 1.9 The Matrix of a Linear Transformation 1.10 Linear Models in Business, Science, & Engineering Focus on engineering examples 7 2.1 Matrix Operations 4 8 2.2 Inverse of a Matrix 9 Review/Test #1 5 10 2.3 Characterizations of Invertible Matrices 11 2.7 Applications to computer graphics (optional) 6 12 3.1 Introduction to Determinants 13 3.2 Properties of Determinants 7 3.3 Cramer s Rule Cramer s Rule only (optional) 14 5.1 Eigenvectors and Eigenvalues 15 5.2 The Characteristic Equation 8 5.3 Diagonalization 16 5.4 Eigenvectors and Linear Transformations 5.5 Complex Eigenvalues 17 5.6 Discrete Dynamical Systems 9 18 Review/Test #2 19 7.4 Basic Theory of Systems of 1 st order Equations 10 7.5 Homogeneous Linear Systems w/ Constant Coefficients 20 7.6 Complex Eigenvalues 7.7 Fundamental Matrices 21 7.8 Repeated Eigenvalues 11 22 10.1 Two-Point Boundary Value Problems 23 10.2 Fourier Series 12 24 10.3 The Fourier Convergence Theorem 25 10.4 Even and Odd Functions 13 26 10.5 Separation of Variables; Heat Conduction 27 10.6 Other Heat Conduction Problems 14 28 Review/Test #3 29 10.7 The Wave Equation 15 30 10.8 Laplace s Equation (optional at OSU) 16 Final Exam