# Math 432/532

## Winter 2020

The problems in parentheses are optional for undergraduates, required for graduate students.
- Syllabus.
- Homework 1 (due 1/10): §1.1 #7, 10, 11, (18). Solutions: pdf, tex.
- Reading 1 (due 1/12): §§1.1–1.3.
- Homework 2 (due 1/17): §1.2 #5, 7, 8, (12). Solutions: pdf, tex.
- Reading 2 (due 1/19): §§1.4, 1.5.
- Homework 3 (due 1/24): §1.3 #5, 8, (10). Solutions: pdf, tex.
- No reading for 1/26.
- Homework 4 (due 1/31): §1.4 #1, 12. §1.5 #2, (6). Solutions: pdf, tex.
- Reading 3 (due 2/2): §§1.6, 1.7 through the first paragraph of page 41. (Leave “Let us illustarte…” for next week.)
- Homework 5 (due 2/7): §1.5 #8. §1.6 #1, 2. Solutions: pdf, tex.
- Midterm 1, solutions, tex.
- No reading for 2/9.
- Homework 6 (due 2/14): §1.6 #7, 9. §1.7 #4, 5. Solutions: pdf, tex.
- Problems on the space of matrices (532 only, due 2/28): pdf, tex.
- Reading 4 (due 2/16): Finish §1.7.
- Homework 7 (due 2/21): §1.7 #3, 13, 14,
~~15~~. Solutions: pdf, tex.
- Midterm 2, solutions, tex.
- Reading 5 (due 3/1): §1.8.
- Homework 8 (due 3/13): §1.8 #2, 6, 7, 8, 9,
~~(14)~~.

Additional problem for 532 students:
Let *m* ≤ *n*, and let *f*: *X* →
**R**^{mn} be a smooth function, where we view the
target as *m×n* matrices. Suppose that for all *x ∈ X*,
the matrix *f(x)* is surjective. Show that there is a smooth map
*g*: *X* → **R**^{nm}, where we view the
target as *n×m* matrices, such that *f(x) g(x)* = 1, where
on the left-hand side we mean matrix multiplication and on the right we mean
the *m×m* identity matrix. (Detailed hint given via email, using
a partition of unity.)
- Take-home final exam: pdf, tex. Optional, due to coronavirus disruptions.