This page will be frequently edited as the semester progresses. Ideally all (and certainly most) of what goes here will be as a consequence of class discussions. "Hefferon" refers to the course text, Linear Algebra (3rd Ed.) by J. Hefferon. See the resources link on the left to access it.

Read the preface and One.I of Hefferon (i.e. everything up to page 34).

Here's the problem we talked about in class today:

- Define a set of lines through the origin in to be equiangular if there is a common angle \( \theta \) such that any pair of lines in the set meets at angle \( \theta \). What is the largest possible set of equiangular lines in \( \mathbb{R}^3 \)? What about \( \mathbb{R}^d \)?

We saw that the answer in \( \mathbb{R}^2 \) is 3, and that the answer in \(\mathbb{R}^3 \) is at least 3.

Three meetings; come to as few or as many as you like.

- 8.30am, Dining Hall, Sage. Get access to Sage via the links on the Resources page and maybe mess with it a bit in advance.
- 1.30pm, Sci217, Vectors. Read Hefferon One.II and come with questions, comments, requests for more examples, etc.
- 2.10pm, Sci 217, LaTeX. Get access to LaTeX via the links on the Resources page and maybe mess with it a bit in advance.

Read One.III of Hefferon. By this point you should also have tried a variety of exercises. As well as making sure the material is clear, we'll talk more about constructing your first portfolio. I'll also give you the big picture overview for Chapter 2, but no need to prep anything on this.

We'll spend this time getting your first portfolios into shape. If there's time, we'll also talk some more about vector spaces (but, again, no need to prep on this yet).

An idea for a portfolio question. Here's something that went through my head on the walk to campus this morning:

- A good way to get equiangular lines is to take diagonals of very symmetrical shapes. We now understand more about \(n\)-cubes and they are very symmetrical. Do the diagonals of an \(n\)-cube give a set of equiangular lines? If not, what goes wrong?

I'm pretty sure that something does go wrong, because if it works perfectly then I think we can beat the known upper bound in higher dimensions. And it's much less likely that we're going to overturn a batch of established math than that the idea doesn't work. So, what goes wrong? To get you started, and move everything into Linalgland, you can think of the \(n\)-cube as being the region in \(n\)-dimensional space with "corners" at all \(n\)-dimensional vectors that consist only of 0s and 1s (or, if you prefer, only -1s and 1s). What vectors represent the diagonals? Take some dot products of these to see what is happening. It might be a good idea to start in 2 or 3 dimensions where the geometry is clearer.

Two.I and Two.II (pp. 78-113).

Hard Problems and Proving Stuff: Mathing like a Pro.

Rest of Chapter 2 (except for the topics). Come ready with places you found difficult or topics you'd like to see more theory/examples for.

NO CLASS: Community Service Day.

Portfolios: Finishing touches (inc. help with LaTex, Sage, hard problems and anything else).

Chapter 3 kick-off. No prep required (although you've been working on your portfolios, of course).

More Chapter 3. Get stuck into the reading and see how far you get.

Last theory day for Chapter 3. Note that we're only going as far as the end of Three.IV (i.e. p. 253). Have read as much as possible of this.

Hendricks Days: No class.

Portfolio tuning. Due tomorrow; come to class with any difficulties you're having.

Gram Schmidt and determinants

Determinants continued, plus some bootcamp time.

Intro to complex numbers and to eigenvalues and eigenvectors

Bootcamp time and come with first thoughts about project topics.

Diagonalisation (5.II in Hefferon)

- More diagonalisation: Fibonacci numbers. No (additional) prep needed.
- Project check-in

- Nilpotence and Jordan Form overview: end of communal theory for the class.

- Project and Portfolio check-in.

- Project and portfolio again. By the end of class today, you should have an abstract and an external evaluator in mind.

NO CLASS: Thanksgiving.

- Project and portfolio: last class tiem to work on these.

- LAST MINUTE SWITCH: Presentations TODAY. Check your email for more details.

- Guest Speaker: Devin Willmott '11 on Machine Learning and Linear Algebra (and Life after Marlboro).

https://cs.marlboro.college /cours /fall2018 /linear_algebra /prep

last modified Sun September 8 2024 6:51 am

last modified Sun September 8 2024 6:51 am