What is the true nature of linearity?

[FredericGos]

Hello,

I'm kind a new to this forum. Been lurking here for a while and got some thought that I'd like to get some advice on. But since I'm new, I better present myself a bit.

I'm a programmer since my youth (age 42 now), and never got a formal degree in anything. Nevertheless, I've always been reading a lot and still doing it, but my focus has changed a lot the latest 4-5 years. Before that, I mostly read IT books about all sorts of stuff in software design, programming, 3D maths for graphics, etc. Today, I have drifted towards reading almost entirely about math & physics. 4-5 years ago, I saw some documentary on string theory and thought that this would be the coolest thing in the world to try to understand... I thought to myself, in my endless naivte, that I would like to understand some of this stuff before i die. :)

I went to amazon and ordered:

http://www.amazon.com/dp/0521357527/?tag=pfamazon01-20

hahaha, well. I opened the book and read the first page... o0

Off course, pretty soon I realized that the body of knowledge needed to be able to understand that page is big. But, hey, I have got the rest of my life, so I just began buying the books backwards recursively if you know what i mean... (i love books). First thing was to study QFT, yeah right, then QM & relativity. Relativity made a lot of sense but my understanding of the tools needed (tensors) was void. Linear algebra was next. That quickly made a lot of sense, even though its a pain. I also sidetracked into QM and quickly found out I had to do EM first, wave math, etc. That also spawned an interest in functional analysis etc. Off course I also soon found out that my calculus & PDE's was also hopeless, so I studied that. back to relativity and differential geometry soon pops up, and you are soon led to fluid dynamics etc. That about where i am by now... I've also side tracked into a myriad of other fields (no pun intended) like abstract algebra, noneuclidian geometry, topology, numerical methods, chaos and more.

As you probably have guessed by now, I'm NOT a scientist. When I say I've studied that, I mean I am (still) studying that, in a constant bouncing back and forth between different disciplines and different levels of complexity. This is the way I work. I prefer to look at it all, increase my knowledge of it all, instead of taking it a step at a time. I know also that this is silly if one is thinking about efficiency, but this is a hobby for me and my ADHD is getting in the way of the linear path.

Which brings me to my question. Maybe its silly, but that is the price to pay for the lack of exactness in my approach and this might be something really fundamental and you will shake your heads and all ;)

1) What does linear mean?

I know about the superposition property and homogeneity conditions for a linear thinggy.
When we use the line in the plane (to illustrate homogeneity?), is this just an analogy that shouldn't be taken literally? The reason I'm asking, is that it seems to me that this is relative to the geometry? What is a line? Can a curve on some manifold by considered linear? A geodesic? Doesn't it all depend on the metric? Is it just because the term 'linear' is bound to the usual euclidean xyz geometry in the mind of us all?

Maybe there is a deeper level of knowledge about the linear/nonlinear world. I cannot help but think that this dichotomy must be fundamental. Is there some literature out there that focuses on this? What books should I read to understand the concept of linearity?

Have a nice evening.

/Frederic

 

[Tac-Tics]

1) What does linear mean?

I know about the superposition property and homogeneity conditions for a linear thinggy.
When we use the line in the plane (to illustrate homogeneity?), is this just an analogy that shouldn't be taken literally? The reason I'm asking, is that it seems to me that this is relative to the geometry? What is a line? Can a curve on some manifold by considered linear? A geodesic? Doesn't it all depend on the metric? Is it just because the term 'linear' is bound to the usual euclidean xyz geometry in the mind of us all?

Linear can mean different things in different contexts. And there's a lot of contexts which can appropriately be called "linear".

First, there are lines in Euclidean geometry. They are straightish.

If you have a cartesean plane, a graph which is a straight line is called a "linear function". That is any function of the form f(x) = mx + b for constants m (the slope) and b (the y-intercept).

If you are talking linear algebra, linear means something different. A linear map or a linear transformation is a function where aT(x) = T(ax) and T(x+y) = T(x) + T(y) for any scalar a and vectors x and y. These functions are very similar to linear functions in the above paragraph, but "b" in mx+b is always 0 in this case (to preserve multiplication with scalars).

A linear combination of vectors is a sum of vectors multiplied by scalars. For example, av + bu + cw = p. Linear differential equations are a spin off of this, where for a function f, we have af + bf' + cf'' = d.

Then there are geodesics and manifolds. I don't know crap about these, except they aren't necessarily "straight" in the conventional Euclidean sense. However, these are hopelessly unrelated to the above definitions for linear functions and linear maps.

Not every book is going to be equally helpful in your understanding. If one book is too confusing, move onto another. And there's no need to jump the gun and study string theory. There are many equally interesting theories with much greater practicality in classical physics.

 

[Crosson]

When we solve linear equations, and imagine the solutions as points or lines in a finite dimensional vector space, then the set of solutions to a homogenous equation with infinitely many solutions can be represented as a line (or a plane) through the origin. The reason we do not need to involve curved geometries is that (1) a vector space has no concept of length and (2) any finite n-dimensional vector space can have a concept of distance defined in a standard way so that it is isomorphic to euclidean n-space.

The entire concept of linearity is about a restricted class of functions and maps. In comparison, the class of analytic functions is highly restrictive , but all analytic functions f(z) can be expressed as a power series and only the ones for which all higher powers of z vanish are called linear, they are all of the form f(z) = a + b z.

Let x,y be elements of a vector space, then T is a linear operator if T(x+y) = Tx + Ty, if a is a scalar we have T(ax) = aTx. At first this seems unrelated to the definition above, but it is easy to prove that an analytic mapping T(x) cannot have any higher order terms in its power series then the linear ones i.e. T(x) = a + b x for scalars a,b.

In differential equations, we say that an equation is linear when it can be represented as a linear operator, defined on an appropriate vector space of functions.

 

[FredericGos]

Thanks for your answers. :)

I think I might have formulated my question poorly. I am aware of the definitions given and think I understand them. Let me try to rephrase.

Lets say we have a metric space and choose a metric that produces a curved geometry. If a solution set in that space forms a curve and we map this to an euclidean space, the result might not be a line even though the function used can still be linear? Or Am I delusional? ;)

The reason I'm thinking about this is because I would like to understand why it is so hard to deal with nonlinear systems. Intuitively I can understand that it must be hard, but if the last paragraph made sense, can't we just map to another space where things become linear?

Frederic