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

### Schedule

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

### Homework

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]

### Handouts

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

### Textbooks

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