Non linear maps between two vector spaces

Question

So, in linear algebra I learned the concept of linear transformation and now I want to learn more about non linear maps between two vector spaces and the mathematical tools that help us understand and analyze them. I've searched it on google but I could not find anything of use. What are the things that I should learn, concepts I should search, books I could read etc. about this topic?

Answer 1

One of the key concepts of modern mathematics is that one of the best and most fruitful ways to study things is by considering structure and functions that preserve that structure.

Vector spaces, for example, are sets with structure: you have addition with a bunch of properties, scalar multiplication with a bunch of properties. You could just take a vector space and stare at it until you figured out interesting things about it. But a better thing to do is to consider all the different ways that you can take that vector space, and either map it to other vector spaces, or map from other vector spaces to it. But not with arbitrary maps. Rather, you want to consider maps that take into account that you are working with vector spaces.

What does it mean for a function $f\colon V\to W$ to “take into account” that you are working with vector spaces? Well, you can add things both in $V$ and in $W$, you can multiply by scalars both in $V$ and in $W$. So you want your map to “keep track” of these operations. This means asking that

1. The image of a sum be the sum of the images: $f(v_1+v_2) = f(v_1)+f(v_2)$; and
2. The image of a scalar multiple be the scalar multiple of the image: $f(\alpha v) = \alpha f(v)$.

Voila! That means linear transformations. So, if you want to think about $V$ as a vector space, then you want to consider linear transformations, and not random functions.

That’s why you aren’t finding much material on considering functions that are not linear between vector spaces. Because, if the function is not linear, then the fact that they are vector spaces is utterly irrelevant.

Now, the same set may have different structures, and so you may be interested in studying some of the other structures. The real numbers are a vector space (for example, over themselves, or over $\mathbb{Q}$). But they are also a topological space; in topology, what you are interested in is functions that respect the notion of “closeness”... which turns out to be the functions that are continuous. And so, you may want to study continuous functions into and out of $\mathbb{R}$, relative to other topological spaces. The same is true for $\mathbb{R}^n$, which also has a structure of a topological space.

Or, in addition to being a topological space, $\mathbb{R}^n$ also has a structure related to differentiability (as in Calculus). The study of maps that respect the differentiable structure is the province of Differential Geometry.

So if you want to look at maps between certain vector space that are not linear, what you really want to do is forget they are vector spaces and find some other structure they have, and look for the maps that respect that structure.

There are some limited exceptions to the above. For example, some vector spaces have an inner product, which allows you to define a distance between vectors. There is the notion of “rigid motion”, which are maps between vector spaces with an inner product that are not necessarily linear, but respect the distance (so that the distance between $x$ and $y$ is the same as the distance between $f(x)$ and $f(y)$); in this situation, there is a connection between these functions and the vector space functions (you can express rigid motions in terms of two simpler types of functions, one of which is a linear transformation). But generally, once you drop the requirement that your functions respect the structure you have, you may as well forget about that structure because it will not play a role in studying those functions.