Hacker News Reader

Comment by CWuestefeld | Hacker News Reader

I simply cannot _recall_ anything unless I understand it at least superficially

Me too. That's why the wheels came off my math skills when I got to differential equations. Everything up to that point I'd been able to form a picture of in my head. For example, calculus is just Newtonian mechanics, so that's got a clear analog in the real world.

But Diff EQ was just too abstract. I was eventually able to pass it with the help of a couple friends pumping me full of formulas and procedures, but I never really understood, and so I never could really apply it on my own.

View on HN

I took a linear algebra class and they explained nothing. Sure they'll prepare you for their tests, but otherwise their material was way too abstract. Not until I watched a lecture by Gilbert Strang on iTunes U did I finally understand what an eigenvalue/vector is and what the hell the matrices represent.

So, enlighten us: what is an eigenvalue, and what do matrices represent?

So mcherm, I'm pretty sure you don't need any enlightenment on this, but I'll try to peer teach this to whom it may help.

So let's say we have a vector which is represented by the matrix A of any dimension (any # of columns, each column a vector). If we multiply this matrix A with an arbitrary unit vector (considering just the direction in the given dimensions) B, then we'll have a certain set of possible resulting vectors (A X B). The eigenvectors are whenever the same unit vector B lines up in direction with (A X B)--where both vectors are dependent on B, and the eigenvalues are the corresponding scalars which must multiply the unit eigenvector to match the resulting A X B vector. The eigenvector does not have to be a unit vector, but since any multiple of the eigenvector can still be a valid eigenvector, you probably just want the unit vector or at least the lowest whole number reduction. If you reduce the eigenvector, the eigenvalue must be changed too.

In equation form: Ax=Ωx Where A is a matrix you are trying to transform, and x is the possible eigenvector, and Ω is the eigenvalue.

In graphical form (this was what really did the trick): http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/Tools... (Demo #1)

And the matrices are simply a different form of a set of equations while still preserving their relative properties. The matrix form allows you to easily manipulate the original set of data with matrix operations. I guess the matrix can also just represent a set of column vectors (which are just a direction in a space with a magnitude) too.

Let me know if I got anything wrong.

I was in that boat for a while being both of a similar mind and in a similar situation. I got over it by the simple work of being absolutely pulled through the ringer by a diabolical prof on Abstract Vector Spaces.

(btw: I love that professor today. He was just underhanded enough to do what we all needed.)

I'm utterly convinced that pure math needs to be taught at a high level for anyone in advanced science/engineering. It hurts for a while, and then you look back and everything is clear, simple, and bright shining.

I've also heard this just keeps happening if you keep on with math.

I have had the same experience in a lot (all?) of my math classes. I never really understood integrals and differentiation until I saw them come up in a physics class. Once I saw the practical application of those things, I didn't have much of an issue in actually solving them. That's the primary reason why I hate math courses. I know there are practical applications to everything I'm learning, but I just can't see them.

When using math for real world applications, you essentially have three steps. First translate the problem into a math problem. Then manipulate the math problem as needed, then finally convert back the real world problem. But in most classes, the second step is the only part that is covered. This is the part where most people have problems, since people are wired to deal with the world around them, not some abstract math world. Trying to ground the math concepts to real world situation is difficult and complicated, which is probably why this step is skipped.

Differential equations often map to physical scenarios - waves, heat diffusion, etc.

Very true, they are incredibly useful. But as taught in most class rooms the application is far removed from the instruction. A differential equations class is often not much more than showing some techniques and then assigning numerous quantative problems that use those techniques.

So, taking a differential equations class by itself can make it seem abstract and with limited connection to the physical world.

Reply on news.ycombinator.com