| Spring 2023 MAT 315/526: Advanced linear algebra | ||
| Schedule | TTh 3:00-4:20pm Earth & Space 69. The first four lectures meet with MAT310 in Engineering 143 | |
| Instructor | Robert Hough | |
| Office hours | F 9am-11am in Math Tower 4-118, 6-7pm in MLC. | |
| Description | Finite dimensional vector spaces over a field, linear maps, isomorphisms, dual spaces, quotient vector spaces, bilinear and quadratic functions, inner products, canonical forms of linear operators, multilinear algebra, tensors. This course serves as an alternative to MAT 310. It is an intensive course, primarily intended for math majors in Advanced Track program. This course is offered as both MAT 315 and MAT 526. | |
| Recitation | Wednesdays from 11:45am-12:40pm in Physics P113 with Yu Xiao. Weekly homework is due each week in recitation. | |
| Textbook | Sheldon Axler. Linear algebra done right. Springer (2015). | |
| Final exam | Tuesday, May 9 from 2:15-5pm in the usual classroom. | |
| Grading | The course grade is based upon homework 20%, two midterms 20% each and final exam 40%. | |
Syllabus/schedule (subject to change)
| Tues 1/24 | 1. | Definitions regarding vector spaces | Axler 1.A,1.B | Homework 1: due 2/1 p.11 #1, 10, 11 p.17 #3, 6 p.24 #1, 3, 11, 12 |
| Thurs 1/26 | 2. | Subspaces | Axler 1.C | |
| Tues 1/31 | 3. | Span and linear independence | Axler 2.A | Homework 2: due 2/8 p.37 #5, 7, 17 p.43 #5, 7, 8 p.48 #3, 6, 15 |
| Thurs 2/2 | 4. | Bases and dimension | Axler 2.B,2.C | In class quiz to determine enrollment in 310/315 |
| Tues 2/7 | 5. | The vector space of linear maps, null space and range | Axler 3.A,3.B | Homework 3: due 2/15 p.57 #8, 13 p.67 #5, 19, 22 p.78 #3, 6 p.88 #5, 10, 14 |
| Thurs 2/9 | 6. | Matrices, isomorphisms | Axler 3.C,3.D | |
| Tues 2/14 | 7. | Products and quotients | Axler 3.E | Homework 4: due 2/22 p.99 #4, 5, 6, 12, 17 p.113 #5, 9, 14, 30, 34 |
| Thurs 2/16 | 8. | Duality | Axler 3.F | |
| Tues 2/21 | Midterm 1 Practice midterm. | Homework 5: due 3/1 p.129 #2, 5, 8, 10 | ||
| Thurs 2/23 | 9. | Polynomials | Axler 4 | |
| Tues 2/28 | 10. | Invariant subspaces and eigenvectors | Axler 5.A,5.B | Homework 6: due 3/8 p.138 #3, 5, 10, 20 p.153 #1, 5, 10 p.160 #1, 7, 13 |
| Thurs 3/2 | 11. | Eigenspaces and diagonalization | Axler 5.C | |
| Tues 3/7 | 12. | Inner products and norms | Axler 6.A | Homework 7: due 3/22 p.175 #5, 6, 9, 12, 20 p.189 #2, 8, 12, 13, 14 |
| Thurs 3/9 | 13. | Orthonormal bases | Axler 6.B | |
| No class - Spring Break 3/13-3/19 | ||||
| Tues 3/21 | 14. | Orthogonal complements and minimization | Axler 6.C | Homework 8: due 3/29 p.201 #1, 4, 5, 8, 11 p.214 #5, 12, 14, 17, 18 |
| Thurs 3/23 | 15. | Self-adjoint and normal operators | Axler 7.A | |
| Tues 3/28 | 16. | The spectral theorem | Axler 7.B | Homework 9: due 4/5 p.223 #5, 6, 12 p.231 #4, 6, 8, 13 p.238 #3, 10, 18 |
| Thurs 3/30 | 17. | Positive operators, polar decomposition and singular value decomposition | Axler 7.C,7.D | |
| Tues 4/4 | 18. | Generalized eigenvectors and decomposition of an operator | Axler 8.A,8.B | Homework 10: due 4/12 p.249 #5, 7, 17, 18 p.259 #3, 4, 9, 10 |
| Thurs 4/6 | 19. | Characteristic and minimal polynomial, Jordan form | Axler 8.C,8.D | |
| Tues 4/11 | 20. | Complexification | Axler 9.A | Homework 11: due 4/19 p.267 #5, 6, 10, 11 p.274 #4, 5, 8 p.285 #7, 15, 19 |
| Thurs 4/13 | Midterm 2 | |||
| Tues 4/18 | 21. | Operators on real inner product space | Axler 9.B | Homework 12: due 4/26p.294 #1, 3, 7 p.304 #1, 4, 7, 10, 18, 20 |
| Thurs 4/20 | 22. | Trace | Axler 10.A | |
| Tues 4/25 | 23. | Determinant | Axler 10.B | Homework 13: due 5/3 p.330 #1, 2, 6, 12 |
| Thurs 4/27 | 24. | Tensors, pull-back | Multilinear algebra | |
| Tues 5/2 | 25. | Alternating forms, wedge product | Multilinear algebra | |
| Thurs 5/4 | 26. | Orientation, signed volume | Multilinear algebra |
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