{"id":3408,"date":"2023-06-09T13:39:04","date_gmt":"2023-06-09T20:39:04","guid":{"rendered":"https:\/\/dornsife.usc.edu\/mathematics\/?page_id=3408"},"modified":"2025-11-29T13:10:57","modified_gmt":"2025-11-29T21:10:57","slug":"math-235-linear-algebra-applications","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/mathematics\/math-235-linear-algebra-applications\/","title":{"rendered":"MATH 235 – Linear Algebra and Applications"},"content":{"rendered":"\n\n \n \n\n\n\n\n\n\n\n
\n\n \n \n
\n\n \n

Math 235 does not have a common final exam.
\nIf evaluating an external course for transfer credit, please see the comments at the bottom of the page.
\n<\/span><\/p>\n

 <\/p>\n

RECOMMENDED TEXTBOOKS<\/strong><\/p>\n

Strang, Introduction to Linear Algebra (6th ed.)<\/a>
\nFriedberg, Insel, Spence, Linear Algebra (5th ed.)<\/a>
\nLay, Lay, McDonald, Linear Algebra and Its Applications (5th ed.)<\/a><\/p>\n

 <\/p>\n

 <\/p>\n

MEASURABLE OBJECTIVES<\/strong><\/p>\n

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. Identify several different inner products and use them to determine distance and angle in a given vector space.<\/li>\n
  13. Understand orthogonality and compute components and projections of vectors.<\/li>\n
  14. Apply techniques for finding eigenvalues and eigenvectors of square matrices and understand their significance in various applications.<\/li>\n
  15. Use diagonalization to simplify and analyze complex systems and transformations.<\/li>\n
  16. Identify and use the Singular Value Decomposition (SVD) of a matrix.<\/li>\n<\/ol>\n

     <\/p>\n

     <\/p>\n

    ADDITIONAL APPLICATIONS<\/strong><\/p>\n

    Math 235 should reserve approximately 2-3 weeks or 6-9 lectures for concrete applications from their chosen textbook.\u00a0 Individual instructors have wide freedom here.\u00a0 Good choices of applications might include:<\/p>\n

     <\/p>\n

      \n
    • Computing techniques in Python, R, or MATLAB.<\/li>\n
    • Orthonormal bases and the Gram-Schmidt algorithm<\/li>\n
    • Least Squares Approximations<\/li>\n
    • Types of matrices such as orthogonal matrices, unitary matrices, positive-definite matrices, etc.<\/li>\n
    • Additional standard forms like LU-factorization or Jordan Canonical Form<\/li>\n
    • Markov Chains<\/li>\n
    • Quadratic forms<\/li>\n
    • Optimization<\/li>\n
    • Applications to image compression<\/li>\n
    • Applications to networks and linear programming<\/li>\n<\/ul>\n

       <\/p>\n

      When evaluating external courses in transfer, a course that covers just the measurable objectives 1-15<\/strong> listed above (excluding SVD) may substitute for Math 235 and earn 3<\/strong> units.\u00a0 This is common for courses taken in quarter systems.\u00a0 A course that contains additional applications<\/strong> as described here may earn 4<\/strong> units.<\/p>\n

      \u00a0<\/strong><\/p>\n

       <\/p>\n

       <\/p>\n

       <\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":328,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-3408","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"\nMATH 235 - Linear Algebra and Applications - Department of Mathematics<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/dornsife.usc.edu\/mathematics\/math-235-linear-algebra-applications\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"MATH 235 - Linear Algebra and Applications - Department of Mathematics\" \/>\n<meta property=\"og:url\" content=\"https:\/\/dornsife.usc.edu\/mathematics\/math-235-linear-algebra-applications\/\" \/>\n<meta property=\"og:site_name\" content=\"Department of Mathematics\" \/>\n<meta property=\"article:modified_time\" content=\"2025-11-29T21:10:57+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/dornsife.usc.edu\/mathematics\/math-235-linear-algebra-applications\/\",\"url\":\"https:\/\/dornsife.usc.edu\/mathematics\/math-235-linear-algebra-applications\/\",\"name\":\"MATH 235 - 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