General information | Course description | Lecture notes | Assignments | Homework schedule | Grading | Links | Fine print

**Lectures:** Monday and Wednesday, 13:25 – 14:40, Physics Building 227

**Text:** *Linear Algebra and its Applications*,
by Peter D. Lax, second edition

**Secondary texts:**

- Climenhaga: Lecture notes for Advanced Linear Algebra
- Hefferon:
*Linear Algebra* - Treil: Linear Algebra Done Wrong
- Cherney, Denton, and Waldron: Linear Algebra
- Cornell: Honors Linear Algebra
- Kristopher Tapp:
*Matrix Groups for Undergraduates*(book; not electronic) - G. W. Stewart and Ji-guang Sun:
*Matrix Perturbation Theory*(book; not electronic) - Gustafson: Advanced Topics in Linear Algebra (not a full text, but a few pages of notes)

**Name:** Professor Ezra Miller
**Address:** Mathematics Department,
Duke University, Box 90320,
Durham, NC 27708-0320
**Office:** Physics 209
**Phone:** (919) 660-2846
**Email:** *ezramath.duke.edu*
**Webpage:**
*https://math.duke.edu/people/ezra-miller*
**Course webpage:** you're already
looking at it... but it's
*https://services.math.duke.edu/~ezra/403/403.html*
**Office hours:**
Monday 14:40 – 15:55 &
Wednesday 16:00 – 17:00, in Physics 209

- abstraction
- approximation
- variation
- positivity
- convexity
- application

Students will continue to develop their skills in mathematical exposition, both written and oral, including proofs.

**Course content:**

- basic algebra: arbitrary fields, quotients, exact sequences, duality
- grassmannians and flags
- classical matrix groups
- Hermitian, positive (semi-)definite, and normal matrices
- principal component analysis (PCA) and singular value decomposition (SVD)
- convexity
- normed linear spaces
- perturbation of eigenvalues
- Perron-Frobenius theory: matrices with positive entries
- tensors and exterior algebra

- Lecture 1: basic algebra of groups, arbitrary fields, isomorphisms
- Lecture 2: quotients, exact sequences, homology
- Lecture 3: Euler characteristic, duality, complex numbers, Hermitian inner products
- Lecture 4: Jordan form, Fundamental Thm of Algebra, direct sum, minimal polynomial, Cayley-Hamilton Thm
- Lecture 5: finish Jordan form; grassmannians as quotients
- Lecture 6: manifolds, e.g. grassmannians
- Lecture 7: flag varieties, metric spaces, isometry (e.g., of spheres)
- Lecture 8: isometries of real inner product spaces, unitary matrices
- Lecture 9: upper-triangular representations, spectral theorem, normal operators, positive (semi)definite operators
- Lecture 10: singular values, Schmidt decomposition, singular value decomposition (SVD)
- Lecture 11: ellipsoids, operator and Frobenius norms, principal component analysis (PCA)
- Lecture 12:
(midterm review), tangent space, Lie algebra as tangent space, computations for
GL
_{n}, SL_{n}, O_{n} - Lecture 13: (midterm review), matrix exponentials, Lie groups as manifolds
- Lecture 14: convex set, polytope, interior, boundary, separating hyperplane
- Lecture 15: supporting hyperplane, extreme point, Krein-Milman Thm
- Lecture 16: Banach spaces: norms, topology, equivalence
- Lecture 17: matrix norms, consistency, general perturbation theorems
- Lecture 18: Bauer-Fike theorem, Gerschgorin theory
- Lecture 19: entrywise positive matrices, dominant eigenvalue, Perron's theorem
- Lecture 20: directed graphs, stochastic matrices, Markov chains, Frobenius theorem
- Lecture 21: multilinear algebra, tensor product, universal property, category
- Lecture 22: alternating operator, exterior power, volume, determinant, functor
- Lecture 23: cross product, Laplace expansion, minor, r-dimensional volume, Plücker coordinate

**Reading assignments**are included at the top of each homework and midterm.- Due dates for the five homework assignments this semester are listed in the table below.
- All assignments, including the midterms, will be
take-home.
**All are due at the start of class time on the due date**. **All solutions you turn in, including midterms, homework, and term projects, must be typewritten.**Communicating your ideas is an integral part of mathematics. In addition to the usual PDF files, I will provide the LaTeX source files for each of the homework assignments as well as each of the midterms. You should feel free to use these as LaTeX templates for your solutions, by simply filling in your responses in those files. I will be happy to answer any questions you might have about LaTeX, although you should ask your classmates first.**Turn in your homework solutions to me by email**. Send at least your .tex file (the grader or I may comment on your TeX usage); if you include your .pdf file as well then it can serve as verification that your system produces the same thing as ours do.**Do not email your midterm solutions to the grader**; email them only to me.**Collaboration on homework is encouraged**, as long as each person understands the solutions, writes them up using their own words, and indicates—on the homework page—who their collaborators were.- In contrast,
**no collaboration or consultation of human or electronic sources—except for the "Text" and "Secondary texts listed above"—is allowed for either of the two exams.** **You must cite sources in your solutions.**If you rely on so-and-so's theorem, then you must state the theorem and tell me where you found it. Be specific: "the dual rank theorem" is not precise; in contrast, "[Climenhaga, Theorem 5.10]" is. Theorems are often known by many names, so I'm likely not to recognize many theorems by names you might attach.

Assignment | Due Date | Problems |
---|---|---|

Homework #1 |
Wed. 1 February | in PDF or LaTeX |

Homework #2 |
Wed. 15 February | in PDF or LaTeX |

Midterm 1 | Wed. 22 February | in PDF or LaTeX |

Homework #3 |
Fri. 10 March | in PDF or LaTeX |

Homework #4 |
Thu. 6 April | in PDF or LaTeX |

Homework #5 |
Fri. 14 April | in PDF or LaTeX |

Midterm 2 | Fri. 21 April | in PDF or LaTeX |

Final project | Friday 5 May, noon |

- 20% Homework
- 20% Midterm #1
- 20% Midterm #2
- 40% Term project

- list of all courses given in the Math Department in Spring 2017.
- Official University calendar for academic year 2016–2017.
- Fundamental University policies concerning academic dishonesty and related matters.

Many thanks are due to Jeremy Martin and Vic Reiner, who provided templates for this webpage many years ago.

The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by Duke University.

*ezramath.duke.edu*

*Fri Apr 14 03:39:22 EDT 2017*

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