MATH 225 – Linear Algebra and Differential Equations

Math 225 does not have a common final exam.

REQUIRED TEXTBOOK

Goode & Annin, Differential Equations and Linear Algebra (4th ed.)

SECTION COVERAGE

The following table lists the minimum set of topics to be included in this course.  The number of lectures listed for each chapter is only a suggestion and will vary across instructors and semesters.

There are typically 42-43 lecture days in a semester, so lecture periods are available for exams.

OPTIONAL TOPICS (time permitting)

• LU-factorization in 2.7 is not used later. Elementary matrices could be introduced with 3.4 instead.
• Sections 3.1 and 3.2 use a formal definition of the determinant by even/odd permutations, which can be omitted.  Augment Sections 3.3/3.4 with specific items from 3.1/3.2 as needed.
• The entirety of Chapter 5 on Inner Products is optional, though important, and at minimum we recommend both the definitions and orthogonality from 5.1/5.2.
• Sections 7.4, 7.5, and 7.6 are all useful applications.
• Chapter 1 has a lot of optional content and applications, with the main definitions and methods appearing in Sections 1.2, 1.4, 1.6, and 1.8.
• Chapter 8 has some optional applications, and Section 8.7 gives an optional method.
• Sections 9.4/9.5 on Vector Differential Equations are optional.
• Section 9.6 may be useful if 8.7 is covered.
• Section 9.8 may be useful if 7.4 is covered.
• Other outside topics, readings, videos, or materials the Instructor deems relevant.

MEASURABLE OBJECTIVES

By the end of this course, students should be able to:

1. Define and explain fundamental concepts in linear algebra, such as vectors, matrices, scalars, and vector spaces.
2. Perform basic operations on vectors and matrices, including addition, subtraction, scalar multiplication, and matrix multiplication.
3. Solve systems of linear equations using various methods, including Gaussian elimination and matrix inversion.
4. Calculate the determinant of a matrix and understand how row operations affect the determinant.
5. Determine the invertibility of a matrix and calculate matrix inverses using Gauss-Jordan technique.
6. Comprehend the notion of vector spaces and be able to determine whether a set of vectors forms a vector space.
7. Analyze the properties of vector spaces, including basis, dimension, linear independence, and span.
8. Comprehend the definition of a linear transformation, and identify whether or not a given function is a linear transformation.
9. Compute change-of-basis matrices and the matrix of a linear transformation.
10. Decide whether or not a given linear transformation is injective or surjective, and compute its kernel and range.
11. Use the Rank-Nullity Theorem to find information about a linear transformation between vector spaces.
12. Apply techniques for finding eigenvalues and eigenvectors of square matrices and understand their significance in various applications.
13. Use diagonalization to simplify and analyze complex systems and transformations.
14. Critically analyze mathematical proofs related to linear algebra concepts and theorems.
15. Define and explain fundamental concepts in differential equations and the classification of equations.
16. Solve some simple first-order equations using standard techniques.
17. Use initial values to identify a particular solution from the general solution.
18. Use the characteristic polynomial to solve homogeneous n-th order linear equations with constant coefficients.
19. Solve non-homogeneous n-th order linear equations and identify the particular and complementary parts of the general solution.
20. Understand that the solutions form a vector space and discuss the dimension of the solution space.
21. Discuss systems of first-order linear differential equations and their notation.
22. Convert an n-th order linear differential equation with constant coefficients into a system of first-order linear equations.
23. Use eigenvalues and eigenvectors to find the general solution to a system of first-order linear equations.