[dalke]

I see that chapter 10, on eigenvalues/eigenvectors, is "Coming soon!".

Eigenvectors never made intuitive sense to me. As with the various decompositions (Cholesky, LU, etc), I could apply the math as algorithms to follow, but never got to the point where felt I could apply them to new problems.

Then again, in practice, I've only needed eigenvectors once since college, and it was more a rote implementation described in a paper. (In other words, don't feel like you should educate me on them here.)

Since you are a TA, don't you get some idea of where your students struggle with linear algebra?

[jeffwass]

Linear algebra (and most calculus) didn't really click for me until I needed to apply it to physics problems.

The general idea started to make sense in mechanics class when I could see matrices are convenient shorthand for solving multiple variables at once, and which behave like 'regular' variables when trying to manipulate them algebraically.

Eigenvalues and eigenvectors didn't make sense until quantum mechanics, where your 'operator' is effectively a matrix and your 'state' is a vector. The allowed observed values are the eigenvalues, and your final state after measurement is the corresponding eigenvector. Wave functions are then an extension of this from discrete space (useful for modelling spins for instance) to the continuum of position space.

[Steuard]

I love eigenvectors. For me, the rest of linear felt like a bunch of rote rules and algorithms for figuring out vaguely-interesting stuff. And then suddenly I find out that there's this amazingly non-obvious but remarkably powerful structure hidden inside matrices that I'd never even been aware of before.

Quantum mechanics is a huge application of eigenvectors, but I also really enjoy things like the "moment of inertia tensor", whose eigenvectors are the natural axes of rotation. Or better still, coupled oscillators, where the eigenvectors give "normal modes" of vibration. (And if you look at coupled first-order differential equations, eigenvectors can tell you all sorts of things about "trajectories" of the solutions. There are great applications of that to things like population dynamics in biology.)

[geomark]

I enjoyed this articled called "Principal Components Analysis for Dummies" that illustrates eigenvectors and eigenvalues. https://georgemdallas.wordpress.com/2013/10/30/principal-com...

[steamer25]

There's also this interactive page for eigen- vectors and values: http://setosa.io/ev/eigenvectors-and-eigenvalues/ (as well as a sibling for PCA: http://setosa.io/ev/principal-component-analysis/)

[dougabug]

The point of an eigenvector is that its direction is stable when you hit it with the given transformation. That is, it's an invariant direction of your transformation. This is analogous to the derivative of an exponential function being a simple multiple of that function. It makes it easier to reason about the action of the matrix, and simpler to represent.

Imagine I have a matrix transformation that takes a given input vector and converts it into a superposition of a thousand other vectors in random directions; that's far harder to reason about than if if just kicked it farther or brought it closer in the same direction.

[cabinpark]

Interesting. I used linear algebra with quantum mechanics where eigenvectors represent quantum state of the Hamiltonian which is how I initially understood them.

And as for TAing, have you ever TA'd? You definitely get a feeling but I can count numerous times when I've stood in front of the tutorial class and asked if there are any questions only to get no response back. I think it's a symptom of first years. I've TA'd calculus as well and I get similar responses. It's very frustrating sometimes.

[Jtsummers]

> You definitely get a feeling but I can count numerous times when I've stood in front of the tutorial class and asked if there are any questions only to get no response back.

When I realized I was understanding subjects better than my classmates, I'd take on the task of asking the "dumb" questions for them. It was partly selfishness, I was tired of answering those questions for them outside of class. But it really did prove helpful to my classmates. It really helped that, when I was sitting with the students, I got to hear them mumbling and grumbling about what they didn't get. So I knew exactly what questions to ask to get the professor/TA to help my classmates.

I never TA'd myself so I have no idea if this would actually work, or at least work consistently. But, if you have a couple students that really seem to be getting the material, you could try talking to them one-on-one and ask them to help you out in this manner.

[dalke]

I have TA'ed. Asking for questions in public doesn't help much. Students rarely like to admit they don't know something.

I was thinking more of when you look at their assignments. There are often multiple ways to approach a problem, and the route chosen can reveal something about one's level of comfort.

It doesn't help that university TAs get almost no training in how to be a TA. At least, I didn't.

[cabinpark]

Yeah I know. It's hard to squeeze information out of them. I actively tell them to please, please come to me if they don't understand something and I have office hours for a reason. Yet no one takes advantage. At a certain point, you can only do so much since I'm a grad student and not a lecturer and my time is finite.

The problem I've found with assignments though is that people copy and cheat. Many times someone will do very well on assignments and then do absolutely terrible on midterms and finals. It's very frustrating. I remember one course where everyone did nearly perfect on the assignments and yet the final and midterm followed the standard bell curve.

[cokernel]

It's worth devoting a portion of classroom time (in my opinion, a substantial portion) to discussion of the topic, perhaps focusing on particular problems and generalizing from there. For example, you might select random students each day to present problems from homework. If you've ever noticed that you learn something better once you teach it to someone else -- well, it works for your students, too.

Moreover, if you create a non-judgmental environment in which people are free to talk about their approaches to problems and get feedback not only from you but from other students as well, then just by watching carefully, you will learn some of the more common gaps in understanding. (Note that some students will not talk in these situations unless forced, but that does not mean they do not benefit from following the discussion.)

If you're anything like I was when I was first TAing courses like this, you might think that if you do this, you won't have enough time to "cover the material". But I put it to you that a lecture that is not absorbed doesn't cover anything.