[neuronic]

That might all be the case. I would simply like a thoughtful introduction to the topic about where this abstraction comes from. What is the idea behind it and what was the motivation for its invention?

After I understand the plain English concept, then give me the math notation and proof and applications.

The way it works today is often to simply leave out the first part and I believe this is why many people find it hard to develop intuition and a real understanding of the concept.

For man it's just there, you memorize it, you apply it, you take the result and simply have faith that it is what you need because some book/ prof said to solve problem X use eigenvalues.

[zornthewise]

The simple idea is that while linear maps(matrices) are generally a combination of rotations, scaling and other stuff in higher dimensions, sometimes they will act on a vector simply by scaling it.

Since scaling is a really easy operation to understand, the space generated by these vectors will be really easy to understand. Note that different vectors can be scaled by different amounts.

Now it often turns out that the space generated like this is actually the whole of the space under consideration and this really simplifies the linear map we started with. Hope that helps.