Linear Analysis I
1. Catalog Description
MATH 244 Linear Analysis I (4) (Also listed as HNRS 244)
Separable and linear ordinary differential equations with selected applications;
numerical and analytical
solutions. Linear algebra: vectors in n-space, matrices, linear transformations,
eigenvalues, eigenvectors,
diagonalization; applications to the study of systems of linear differential
equations. 4 lectures.
Prerequisite: MATH 143 or consent of instructor.
2. Required Background or Experience
Math 143 or equivalent.
3. Learning Objectives
The student should:
a. Develop an understanding of the elementary theory of ordinary differential
equations and
their solutions in the context of separable and first order linear ordinary
differential equations
and recognize situations where these equations arise in selected mathematical
models.
b. Become familiar with the terminology and methods of solution of higher order
linear
differential equations, especially methods for constant coefficient homogeneous
and forced
equations, and understand how these solution methods apply to selected second
order
linear models.
c. Develop an understanding of linear algebra in Euclidean n-space and how
vectors and
matrices are used to solve systems of equations, both algebraic and
differential.
4. Text and References
Edwards/Penney/Greenberg, Differential Equations and Linear Algebra – Custom
Edition for
Cal Poly SLO, Pearson/Prentice-Hall, 2006.
5. Minimum Student Materials
Paper, pencils, calculator and notebook.
6. Minimum University Facilities
Classroom with ample chalkboard space and computer laboratory.
7. Content and Method
Topic | Days |
Chapter 1. First Order Differential Equations | 5 |
1.1 Differential equations and mathematical modeling
1.2 Integrals as general and particular solutions
1.3 Slope fields and solution curves
1.4 Separable equations and applications
1.5 Linear first-order equations
Chapter 2. Mathematical Models and Numerical Methods | 1 |
2.4 Numerical approximations: Euler’s method
Chapter 3. Linear Systems and Matrices | 6 |
3.1 Introduction to linear systems
3.2 Matrices and Gaussian elimination
3.3 Reduced row-echelon matrices
3.4 Matrix operations
3.5 Inverses of matrices
3.6 Determinants
Chapter 4. Vector Spaces | 7 |
4.1 The vector space R^{3}
4.2 The vector space R^{n} and subspaces
4.3 Linear combinations and independence of vectors
4.4 Bases and dimension for vector spaces
4.5 Row and column spaces
4.7 General vector spaces (May be omitted if time is short)
Chapter 5. Linear Equations of Higher Order | 7 |
5.1 Introduction: Second-order linear equations
5.2 General solutions of linear equations
5.3 Homogeneous equations with constant coefficients
5.4 Mechanical vibrations
5.5 Nonhomogeneous equations and undetermined coefficients
5.6 Forced oscillations and resonance
Chapter 6. Eigenvalues and Eigenvectors | 3 |
6.1 Introduction to eigenvalues
6.2 Diagonalization of matrices
Chapter 7. Linear Systems of Differential Equations | 4 |
7.1 First-order systems and applications
7.2 Matrices and linear systems
7.3 The eigenvalue method for linear systems
Total |
33 |
The principal emphasis in Chapter 5 should be on
second-order equations.
Optional topics (only if time permits)
6.3 Applications involving powers of matrices
7.4 Second-order systems and mechanical applications
7.5 Multiple eigenvalue solutions
7.6 Numerical methods for systems
Method
Lecture/discussion and regular homework assignments.
8. Methods of Assessment
Homework assignments, tests, classroom participation and a comprehensive final
examination.