Fall 2018

# Preparation

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.

## Monday Sep 3rd

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.

## Thursday Sep 6th

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.

## Monday Sep 10th

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.

## Thursday Sep 13th

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.

## Monday Sep 17th

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

## Thursday Sep 20th

Hard Problems and Proving Stuff: Mathing like a Pro.

## Monday Sep 24th

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.

## Thursday Sep 27th

NO CLASS: Community Service Day.

## Monday Oct 1st

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

## Thursday Oct 4th

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

## Monday Oct 8th

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

## Thursday Oct 11th

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.

## Monday Oct 15th

Hendricks Days: No class.

## Thursday Oct 18th

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

## Monday Oct 22nd

Gram Schmidt and determinants

## Thursday Oct 25th

Determinants continued, plus some bootcamp time.

## Monday Oct 29th

Intro to complex numbers and to eigenvalues and eigenvectors

## Thursday Nov 1st

Bootcamp time and come with first thoughts about project topics.

## Monday Nov 5th

Diagonalisation (5.II in Hefferon)

## Thursday Nov 8th

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

## Monday Nov 12th

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

## Thursday Nov 15th

• Project and Portfolio check-in.

## Monday Nov 19th

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

## Thursday Nov 22nd

NO CLASS: Thanksgiving.

## Monday Nov 26th

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

## Thursday Nov 29th

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

## Monday Dec 3rd

• Guest Speaker: Devin Willmott '11 on Machine Learning and Linear Algebra (and Life after Marlboro).
https://cs.marlboro.college /cours /fall2018 /linear_algebra /prep