Math 2135 Fall 2019

MATH 2135: Linear Algebra for Math Majors (Fall 2019)

Syllabus and details on accomodations, honor code, etc

Final exam:

Monday, December 16, 1:30-4:00 pm, STAD 135

Office hours:

Tuesday 10-11 am, Wednesday 3-4 pm


Numbers refer to sections in Treil, Linear algebra done wrong, if not indicated otherwise.
  1. 08/26: some applications of linear algebra
  2. 08/28: column vectors over the reals, addition, scalar multiples (1.1), systems of linear equations, augmented matrix,
  3. 08/30: matrix times column vector (1.3.2), row reduction (2.2)
  4. 09/04: (reduced) echelon form, elementary row operations, free variables, solution in parametrized form (2.3)
  5. 09/06: existence and number of solutions of Ax=b (2.3), homogenous and inhomogenous systems (2.6), nullspace of A (2.6)
  6. 09/09: axioms and properties of fields (Halmos 1.1)
  7. 09/11: integers modulo n, axioms of vector spaces over arbitrary fields (1.1)
  8. 09/13: tuples, matrices, sequences, functions as vector spaces (1.1), properties of vector spaces
  9. 09/16: linear combinations, span of vectors (1.2)
  10. 09/18: column space, Ax=b is consistent iff b is in Col A, row space, transpose of matrix
  11. 09/20: properties of subspaces, spans and null space as subspaces (1.7)
  12. 09/23: linear independent vectors (1.2)
  13. 09/25: basis (1.2)
  14. 09/27: Spanning Set Theorem to remove vectors from a spanning set to obtain basis, coordinates relative to a basis (existence and uniqueness)
  15. 09/30: review for midterm [pdf]
  16. 10/02: MIDTERM
  17. 10/04: linear independent sets cannot be bigger than spanning sets, dimension of a vector space (2.3, 2.5)
  18. 10/07: every linear independent set extends to a basis (2.5), basis theorem
  19. 10/09: dim Nul A, dim Col A, linear transformations (1.3)
  20. 10/11: standard matrix of linear maps (1.3), rotation and reflection as linear maps, matrix multiplication (1.5) [notes]
  21. 10/14: properties of matrix products [notes]
  22. 10/16: invertible matrix, computing inverse matrix by row reduction (2.4) [notes]
  23. 10/18: Invertible Matrix Theorem (1.6.2, 2.3.2)
  24. 10/21: matrix of a linear map w.r.t. bases B,C (2.8.2), change of coordinates matrix (2.8.3) [notes]
  25. 10/23: reflection on arbitrary lines in R^2 (1.5.2) [notes]
  26. 10/25: computer graphics, homogenous coordinates for R^2
  27. 10/28: rotation around line in R^3, injective, surjective (see handout below), kernel, range of linear maps (1.7) [notes]
  28. 10/30: coordinate map as an isomorphism from V with dim V = n to F^n (1.6.3) [notes]
  29. 11/01: determinants, cofactor expansion by a row or column (3.1) [notes]
  30. 11/04: review for midterm [pdf]
  31. 11/06: MIDTERM
  32. 11/08: discussion of midterm, determinant by row reduction (3.3)
  33. 11/11: A is invertible iff det A <> 0 (3.3), det AB = det A det B (3.3), determinant as area of parallelogram
  34. 11/13: eigenvalues, eigenvectors, eigenspaces, characteristic polynomial of a matrix (4.1)
  35. 11/15: eigenvalues of triangular matrix, diagonalizable matrices
  36. 11/18: Diagonalization Theorem (4.2.1), basis of eigenvectors
  37. 11/20: dynamical systems, Fibonacci sequence
  38. 11/22: dot product, length of vectors over the reals (5.1.1), orthogonal basis, coordinates via dot product (5.2)
  39. 12/02: orthogonal projection onto a vector and onto a subspace of R^n (cf. 5.3)
  40. 12/04: Gram-Schmidt algorithm for finding orthogonal basis
  41. 12/06: least squares solution of inconsistent systems (5.4)
  42. 12/09: review [pdf] [practice final]
  43. 12/11: review


  1. due 08/30 [pdf]
  2. due 09/06 [pdf]
  3. due 09/13 [pdf]
  4. due 09/20 [pdf]
  5. due 09/27 [pdf]
  6. due 10/04 [pdf] part preparation for midterm 1
  7. due 10/11 [pdf]
  8. due 10/18 [pdf]
  9. due 10/25 [pdf]
  10. due 11/01 [pdf]
  11. due 11/08 [pdf] part preparation for midterm 2
  12. due 11/15 [pdf]
  13. due 11/22 [pdf]
  14. due 12/06 [pdf]


  1. Integers modulo n [pdf]
  2. Functions [pdf]


We will mainly use the following book which is available for free online: Some additional textbooks:

How to succeed in this class

  1. Go to class! It seems obvious, but learning the material in small portions 3 times a week is easier than reading up on it in some book by yourself. Always keep up with the topics. You also get nerdy Math jokes.
  2. Ask questions early and often! If you are not sure about something, ask about it immediately -- no matter whether in class, in office hours, or by mail. Do not assume that you can skip or figure out things later that you do not understand now. If you are missing the basics, you may fall behind and struggle with more complicated concepts later in class.
  3. Do the work! The only way to learn stuff is to try it yourself. Strive to do all the homework assignments. Some will be more challenging than others. If you are stuck on the hard ones, discuss them with colleagues or ask for possible hints in office hours or by mail.
  4. Learn from mistakes! Look at all feedback you get on graded homework, quizzes, exams, etc. Make sure you understand where you went wrong and how to get the correct solution. In particular revise all relevant graded work before exams.
  5. Organize in study groups! Meet with classmates a couple of times a week to discuss lectures and homework. Still write up your solutions to assignments when you are alone, never in a group.
  6. Take advantage of office hours! If you cannot make it to the official hours, ask to meet at some other time. Office hours are an additional resource for you to discuss stuff for which there is no time during class. Come prepared! Try to solve homework problems alone before you ask for help and be ready to explain your thoughts and where you are stuck.

Scientific writing

There is a variety of word-processing software for writing Mathematics. LaTeX is the most widespread. You can use it with many text editors or via some cloud-based service, like Overleaf.