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

Schedule

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]

Homework

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]

Handouts

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

Textbooks

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