Weekly Agenda

Week 1: Linearity, Mathematical "Nice-ness"

• Tue, Apr 2: Introduction to the course, review of three-vectors, what we mean by linear, introduction to basis vectors. Reference: See Section 3 of the 2023 Course Notes ("Basics").

• Thu, Apr 4: Linearity, "nice" transformations (symmetries) as a motivation for tensors, rotations as an example. Notation. 

Homework

Due dates listed first.

• Thu Apr 6. Please review the syllabus. (No submission link, I'll just trust you that you reviewed it)

• Thu Apr. 6: Pre-Class Survey 

• Thu Apr 6: Intro Video (Prof. Tanedo's welcome video)

• Thu Apr. 6: Short homework 1: Summation convention. Submission link.  The short homework should be quick and something that you can easily after class. It is useful feedback for Prof. Tanedo.  (4/3 update: removed a line in the submission link about the explainer video. Thanks Nick M.)

• Friday Apr. 12: Long homework 1: Vector and matrix fundamentals. Submission link. A review of standard operations on three-vectors, finding the inverse of a 3x3 matrix (see short HW2). Update 4/4: homework due date moved to Friday, previously there was a problem 5 which we have moved to the optional extra credit. You are strongly encouraged to attempt it.

Reading 

The following reading suggestions are provided to supplement/complement the lecture material. 

• Singh, Linear Algebra Step By Step. The basics of linear algebra are in chapter 1. Please ignore anything to do with linear equations. This is not a productive perspective for us. A good starting point is chapter 2, which covers Euclidean vectors. Chapter 3 connects to general properties. Unfortunately Singh does not use our index notation.

• Fleisch, A Student's Guide to Vectors and Tensors. Chapter 5 introduces the index notation, though it is a little ahead of where we are right now.

• 2023 Course Notes:   See section 3 (basics), 4 (index notation), and 5 (ket notation). Note: section numbering may get shuffled as the course progresses and the notes are revised. 

• 2022 Course notes:  See the first few lectures.

Week 2: Bases and Rotations

• Lec 3 (4/9): Basis, change of basis. Matrices as the components of a linear transformation. For a good discussion of basis vectors, here see the April 10 save state of the 2024 course notes.

• Lec 4 (4/11): Dual vectors. Rotations and their action on vectors, dual vectors, matrices. Ket notation.

Homework

The explainer assignments are available on our internal sheet.

• Short homework 2 (due Thursday, 4/11): The inverse of a matrix. Submission link.

• [same as last week] Friday Apr. 12: Long homework 1: Vector and matrix fundamentals. Submission link. A review of standard operations on three-vectors, finding the inverse of a 3x3 matrix (see short HW2). Update 4/4: homework due date moved to Friday, previously there was a problem 5 which we have moved to the optional extra credit. You are strongly encouraged to attempt it.

Reading

• I liked that in class we got some questions about the difference between row vectors and column vectors, and especially the question: where does this show up in physics? For a teaser of how this shows up in special relativity, please see this selection from my Physics 165 (Particle Physics) notes.

• Singh: Chapter 2 discusses the properties of R3. Specifically, sections 2.3 and 2.4 discuss linear independence and bases. Chapter 3 generalizes all of these ideas. You can skip the last section of chapter 3 on linear systems

• Feldman: Chapter 4 talks about covariant and contravariant vectors. This is a general study of rotations. Chapter 4.5, in particular, gives some intuition for what a row vector (covariant vector) means. 

• Extra: there is a cute alternative notation to upper and lower indices called bird tracks. You can read about it here:

  • Kim et al. "Boosting vector calculus with the graphical notation" American Journal of Physics 89, 200–209 (2021)

  • Alcock-Zeilinger and Weigert, "Simplification rules for birdtrack operators" J. Math. Phys. 58, 051701 (2017)

Week 3: Bases, Linear Transformations

• Lec 5 (4/16): Tensors as linear transformations, the basis for linear transformations. This is now written up carefully in Section 4 of the April 16 save state of the 2024 course notes. (Better: see the April 23 version which is essentially complete. It includes a section appendix on the Kronecker delta..)

• Lec 6: Metric spaces. Our discussionis summarized in Section 6 of the course notes, April 23 save state.

Homework

The explainer assignments are available on our internal sheet.

• Short homework 3 (due Thursday, 4/18): The inverse of a matrix. Submission link. [4/17: updated submission link, thanks Daniel C.!]

• Long homework 2 (due Friday, 4/26) [Submission link]

• Explainer video 1 (due Friday 4/18) [Submission Link]

Reading

• April 16 save state of the 2024 course notes.

Week 6: Eigenstuff II

Here's the link to the May 7th snapshot of our 2024 course notes. Some highlights: Chapter 10 (still in progress) is on eigenvectors, Chapter 9 is a deep dive into determinants. Chapter 6 summarizes all of our rules for metric spaces.

• Lec 11 (5/7): Introduction to the eigenvalue problem. 

• Lec 12 (no class): No lecture today. Here are some videos in lieu of the lecture:

  • Rotations 1 (13:28)

  • Rotations 2 (17:53)

  • Dual Bases (8:16)

  • Adjoint (17:31)

  • Determinants 1 (23:12)

  • Determinants 2 (10:11)

  • Determinants 3 (35:11)

Homework

The explainer assignments are available on our internal sheet.

• Short Homework 6 (due Thursday, 5/9) [Submission link]

  • (If you didn't fill this out last week: submission of PERTS Ascend survey)

• Long Homework 3 (due Friday, 5/10) [Submission link]

• Explainer Video 2 (due Thursday 5/9) [Submission link] The explainer assignments are available on our internal sheet.

Week 7: function space

This week we dive into complex vector spaces as a segue into function space. This leads us to Fourier series and Fourier transforms. 

• Lec 13: Quick eigenstuff review (featuring: degenerate eigenvalues), quick introduction to function space.

• Lec 14: introduction some formalism: complex space, complex inner product, the adjoint. Introduction to function space. 

Homework

The explainer assignments are available on our internal sheet.

• Short Homework 7 [submission link]

• Long Homework 4 [submission link] Due May 23

• Explainers and peer reviews to be posted (none due this week)

Additional Reading

• A nice interactive demonstration of determinants from the Immersive Linear Algebra online, interactive textbook. 

• Eigenvectors and eigenvalues explained visually.

Week 8: Adventures in Function Space

May 20 snapshot of 2024 Course Notes: includes some discussion of complex vector spaces, Fourier series, Fourier transform, and function space. 

• Lecture 15: The Fourier Series as a metric space

• Lecture 16: No class. Video recordings of a make up lecture will be posted here. My apologies.

Homework

• Short Homework 8 [submission link] Due May 23

• Long Homework 4 [submission link] Due May 24

• PERTs survey 2. Due May 25

Reading

• The May 20 snapshot of our 2024 course notes now has some material on complex vector spaces and function spaces. See Chapters 13 and 14. 

• Last week before one of our classes we talked about Richard Feynman. There is a recent 3-part podcast by Freakonomics about him focusing partly on his physics, but mostly on him as a human being.

Week 9: Fourier Problems

• Lecture 17: Review of eigenstuff. A Separation of variables problem.

• Lecture 18: Separation of variables, peek at spherical harmonics. 

Homework

• Short Homework 9 [submission link] Due May 30

• Long Homework 5 [submission link] Due June 7 [6/3 link updated, thanks Nick M.)

• Explainer Video 3 [submission link] Due June 7; assignments

• Peer Review 3 [submission link] Due June 7; assignments

• PERTS survey 3. Due end of term.

Reading

• The May 20 snapshot of our 2024 course notes now has some material on complex vector spaces and function spaces. See Chapters 13 and 14. 

Week 10: Fourier and Waves

This week we present a few topics that extend our understanding of linear algebra to mathematical tools that you will see in your undergraduate physics career. There are no new homework assignments this week. The material is mainly to build bridges to the mathematical topics you will encounter in upper division physics (and to show why much of it is actually linear algebra). 

• Lecture 19 (notes): From Fourier series to Fourier transforms

• Lecture 20 (notes): Green's functions, Poisson equation, wave equation, diffusion equation.

Homework

• Long Homework 5 [submission link] Due June 7 [6/3 link updated, thanks Nick M.)

• Explainer Video 3 [submission link] Due June 7; assignments

• Peer Review 3 [submission link] Due June 7; assignments

• PERTS survey 3. Due end of term.

Reading

• The May 20 snapshot of our 2024 course notes now has some material on complex vector spaces and function spaces. See Chapters 13 and 14.