Math 2135 Spring 2019

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

Syllabus and details on accomodations, honor code, etc

Final exam:

Sunday, 05/05/2019, 7:30-10:00 pm, ECCR 151

Office hours:

Tuesday 10-11 am, Wednesday 1-2 pm


Numbers refer to sections in Treil, Linear algebra done wrong, if not indicated otherwise.
  1. 01/14: some applications of linear algebra
  2. 01/16: systems of linear equations, augmented matrix, free variables, solution in parametrized form, row reduction (2.3)
  3. 01/18: (reduced) echelon form, elementary row operations (2.3), matrix times column vector (1.3.2)
  4. 01/23: existence and number of solutions of Ax=b (2.3), homogenous and inhomogenous systems (2.6), nullspace of A (2.6)
  5. 01/25: axioms and properties of fields (Halmos 1.1)
  6. 01/28: axioms of vector spaces over arbirtrary fields (1.1)
  7. 01/30: tuples, matrices, sequences, functions as vector spaces (1.1), properties of vector spaces
  8. 02/01: linear combinations, span of vectors (1.2)
  9. 02/04: column space, Ax=b is consistent iff b is in Col A, row space, transpose
  10. 02/06: properties of subspaces, spans and null space as subspaces (1.7)
  11. 02/08: linear independent vectors (1.2)
  12. 02/11: basis (1.2)
  13. 02/13: Spanning Set Theorem to remove vectors from a spanning set to obtain basis
  14. 02/15: coordinates relative to a basis (existence and uniqueness)
  15. 02/18: review for midterm [pdf]
  16. 02/20: MIDTERM
  17. 02/22: linear independent sets cannot be bigger than spanning sets, dimension of a vector space (2.3, 2.5)
  18. 02/25: every linear independent set extends to a basis (2.5), dim Nul A, dim Col A
  19. 02/27: basis theorem, linear transformations (1.3)
  20. 03/01: standard matrix of linear maps (1.3)
  21. 03/04: matrix multiplication (1.5)
  22. 03/06: invertible matrix, computing inverse matrix by row reduction (2.4)
  23. 03/08: Invertible Matrix Theorem (1.6.2, 2.3.2)
  24. 03/11: matrix of a linear map w.r.t. bases B,C (2.8.2)
  25. 03/15: change of coordinates matrix (2.8.3)
  26. 03/18: reflection on arbitrary lines in R^2 (1.5.2)
  27. 03/20: injective, surjective (see handout below), kernel, range of linear maps (1.7)
  28. 03/22: coordinate map as an isomorphism from V with dim V = n to F^n (1.6.3)
  29. 04/01: review for midterm [pdf]
  30. 04/03: MIDTERM
  31. 04/05: discussion of midterm
  32. 04/08: determinants, cofactor expansion by a row or column (3.1)
  33. 04/10: determinant by row reduction, determinant as alternating multilinear form, A is invertible iff det A <> 0 (3.3)
  34. 04/12: det AB = det A det B (3.3), determinant as area of parallelogram
  35. 04/15: eigenvalues, eigenvectors, eigenspaces (4.1)
  36. 04/17: characteristic polynomial of a matrix (4.1)
  37. 04/19: diagonalizing matrices, Diagonalization Theorem (4.2.1)
  38. 04/22: dynamical systems, Fibonacci sequence
  39. 04/24: dot product, length of vectors over the reals (5.1.1), orthogonal basis, coordinates via dot product (5.2)
  40. 04/26: orthogonal projection onto a vector (cf. 5.3)
  41. 04/29: least squares solution of inconsistent systems (5.4)
  42. 05/01: review [pdf]


  1. due 01/25 [pdf]
  2. due 02/01 [pdf]
  3. due 02/08 [pdf]
  4. due 02/15 [pdf]
  5. due 02/22 [pdf], part preparation for midterm 1
  6. due 03/01 [pdf]
  7. due 03/08 [pdf]
  8. due 03/15 [pdf]
  9. due 03/22 [pdf]
  10. due 04/05 [pdf], preparation for midterm 2
  11. due 04/12 [pdf]
  12. due 04/19 [pdf]
  13. due 04/26 [pdf]


  1. Multiplication of matrix by vector [pdf]
  2. Integers modulo n [pdf]
  3. 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.