{"id":1435,"date":"2023-02-15T12:01:00","date_gmt":"2023-02-15T20:01:00","guid":{"rendered":"https:\/\/dornsife.usc.edu\/mathematics\/?page_id=1435"},"modified":"2023-08-05T12:50:33","modified_gmt":"2023-08-05T19:50:33","slug":"math-225-linear-algebra-and-linear-differential-equations","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/mathematics\/math-225-linear-algebra-and-linear-differential-equations\/","title":{"rendered":"MATH 225 – Linear Algebra and Differential Equations"},"content":{"rendered":"\n\n \n \n\n\n\n\n\n\n\n
\n\n \n \n
\n\n \n

Math 225 does not have a common final exam.<\/span><\/p>\n

 <\/p>\n

REQUIRED TEXTBOOK<\/strong><\/p>\n

Goode & Annin, Differential Equations and Linear Algebra (4th ed.)<\/a><\/p>\n

 <\/p>\n

SECTION COVERAGE<\/strong><\/p>\n

The following table lists the minimum<\/em> set of topics to be included in this course.\u00a0 The number of lectures listed for each chapter is only a suggestion and will vary across instructors and semesters.<\/p>\n

There are typically 42-43 lecture days in a semester, so lecture periods are available for exams.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Sections<\/strong><\/td>\nTopics<\/strong><\/td>\nLectures<\/strong><\/td>\n<\/tr>\n
2.1 – 2.8<\/td>\nMatrices\/Systems of Equations<\/td>\n8<\/td>\n<\/tr>\n
3.3, 3.4<\/td>\nDeterminants<\/td>\n3<\/td>\n<\/tr>\n
4.1 – 4.10<\/td>\nVector Spaces<\/td>\n9<\/td>\n<\/tr>\n
6.1 – 6.5<\/td>\nLinear Transformations<\/td>\n6<\/td>\n<\/tr>\n
7.1 – 7.3<\/td>\nEigenvalues\/Eigenvectors<\/td>\n5<\/td>\n<\/tr>\n
<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
1.2, 1.6<\/td>\nFirst order linear Diff Eqns<\/td>\n3<\/td>\n<\/tr>\n
8.1 – 8.4<\/td>\nn-th order linear Diff Eqns<\/td>\n3<\/td>\n<\/tr>\n
9.1 – 9.3<\/td>\nSystems of linear Diff Eqns<\/td>\n3<\/td>\n<\/tr>\n
<\/td>\nTotal<\/td>\n40<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

OPTIONAL TOPICS (time permitting)<\/strong><\/p>\n

    \n
  • LU-factorization in 2.7 is not used later. Elementary matrices could be introduced with 3.4 instead.<\/li>\n
  • Sections 3.1 and 3.2 use a formal definition of the determinant by even\/odd permutations, which can be omitted.\u00a0 Augment Sections 3.3\/3.4 with specific items from 3.1\/3.2 as needed.<\/li>\n
  • 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.<\/li>\n
  • Sections 7.4, 7.5, and 7.6 are all useful applications.<\/li>\n
  • 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.<\/li>\n
  • Chapter 8 has some optional applications, and Section 8.7 gives an optional method.<\/li>\n
  • Sections 9.4\/9.5 on Vector Differential Equations are optional.<\/li>\n
  • Section 9.6 may be useful if 8.7 is covered.<\/li>\n
  • Section 9.8 may be useful if 7.4 is covered.<\/li>\n
  • Other outside topics, readings, videos, or materials the Instructor deems relevant.<\/li>\n<\/ul>\n

     <\/p>\n

    MEASURABLE OBJECTIVES<\/p>\n

    <\/strong>By the end of this course, students should be able to:<\/p>\n

     <\/p>\n

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

       <\/p>\n

       <\/p>\n

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