Math 4031 Linear Algebra


Catalog Description

Three credits. Three hours of lecture per week. Prerequisite: MATH 3151.

Matrix algebra. Determinants. Systems of linear equations. Finite dimensional vector spaces. Linear dependence and independence. Bases. Euclidean spaces. Linear transformations. The matrix of a linear transformation. Similarity transformations. Eigenvectors and eigenvalues. Inner products. Norms. Orthogonal projection. Gram-Schmidt orthogonalization.

2018-2019– Semester II

Miscellaneous Notes

  1. Matrix Multiplication
  2. Elimination and Back Substitution.
  3. The Inverse of a Matrix.
  4. LU Factorization.
  5. Permutations and Transpositions
  6. Reduction to the Identity
  7. Linear Spaces
  8. Linear Independence
  9. Spanning Sets and Independent Sets
  10. Exam I -Practise Questions
  11. Euclidean Space
  12. Lines in Euclidean Space
  13. Gram Schmidt Algorithm
  14. Orientation in $R^2$
  15. The Vector Product
  16. Planes
  17. Linear Equations
  18. Linear Equations – Simple Examples
  19. Subspaces (updated April 27 2019)
  20. The Hermite Normal Form
  21. Matrix of a Linear Map
  22. Dimensions of Kernel and Image 
  23. The Null Space of a Matrix
  24. Finding the Left Nullspace
  25. Determinants.
  26. Determinant of a Direct Product.
  27. Graph Matrices
  28. Circuits
  29. Closure
  30. Affine Spaces in Physics
  31. Distance to an Affine Subspace
  32. Convex Sets
  33. The Centroid
  34. Projection and Least Squares


  1. Homework on Matrices.
  2. Homework on Elementary Matrices and LU Factorization.H
  3. Homework on Geometry


  1. Lang, Serge, Introduction to Linear Algebra, Springer, 1997.
  2. Gilbert Strang, Introduction to Linear Algebra, Ed. 4, 2009.


  1. Gilbert Strang, Introduction to Linear Algebra, Ed. 4, 2009.
  2. David Santos, Linear Algebra Notes, 2010.
  3. Sheldon Axler, Linear Algebra Done Right, Springer, 1997.
  4. Sergei Treil, Linear Algebra Done Wrong.
  5. E. D. Nering, Linear Algebra and Matrix Theory , Wiley, 1970.
  6. I. Satake, Linear Algebra, M. Dekker, Inc., New York, 1975.
  7. G. E. Shilov, Linear Algebra, Dover Publications, New York, 1977.
  8. Lang, Serge, Linear Algebra, Addison Wesley, 1968.
  9. Peter D. Lax, Linear Algebra and Its Applications, 2nd Edition, 2007.
  10. I. M. Gelfand, Lectures on Linear Algebra, Dover 1989 (translation).

Online Tools and Video Resources

  1. Multipurpose Calculator
  2. Online Lectures from MIT by Gilbert Strang.
  3. Linear Algebra Toolkit