Is advanced college math (eg analysis, abstract/linear algebra, topology) supposed to be as intuitive as elementary math?

Question

So I don't know if I'm not smart enough for math, but lately, it seems to me as if some advanced topics are just too unintuitive in my opinion.

For example, I have no idea what eigenvalues, jacobians or manifolds really are, and it's a similar thing with most of abstract algebra (or at least I've never been told how mathematicians came up with these kinds of concepts -I don't even know what motivated the formulation of a matrix-).

And it's not just algebra. Even though I did pretty well (intuition-wise) in one-variable calculus, I have absolutely no idea why the solutions to differential equations make sense, or the notion behind differential operators.

So, I think one of these things is going on here:

a) I'm not smart enough to gain intuition of these concepts on my own;

b) It is complicated, but possible, to understand intuitively these concepts. Problem is this educational system doesn't put too much emphasis on deep comprehension;

c) Nobody knows very well these areas of math, we just apply rules and we're deeply boggled by how far we can go with them.

Which one of these is the case? I could really use some guidance.

Answer 1

My experience is that if you spend enough time both actively and passively (subconsciously) thinking about the material, these mathematical concepts will become as natural as breathing. It is a slow process at first (it took me until my junior year of undergrad to really start to understand what math was really about) but once you have a sufficient experience and knowledge pool to draw from, it becomes second nature.

Some people call this process "mathematical maturity" but the basic principal still holds: Put the time in and the intuition will come. Now, it is very possible that one's capacity for understanding/learning is bounded by raw intelligence limitations but I'd imagine anyone of average intelligence can at least understand calculus at a reasonably deep level. Most people just lack the patience and drive to master the material to the point that it becomes natural.

Being more of the analytical persuasion, I tend to see math as an interplay between three main things (very informal):

1. Mathematical objects and the spaces to which they belong
2. Mappings between these spaces of objects (which often become objects themselves)
3. The structures of these spaces

These concepts will become much more understandable if you first understand which class of objects are you working with, then consider how these objects can be transformed (or how they transform another class of objects) into other mathematical objects. From there, one can consider how these objects behave in relation to one another relative to the set or space in which they reside.

Just as an example that ties all of your "unintuitive" concepts together, consider the linear operator $D:C^{k} \to C^{k-1}$ given by $$ D=\frac{d}{dx} $$ now the only eigenfunction for this operator is $f=e^{\lambda x}$ where $\lambda \in \mathbb{R}$ since we know from calculus that $$ Df=\lambda e^{\lambda x} $$ so $\lambda$ is an eigenvalue for any $\lambda \in \mathbb{R}$. Note that since $D$ is a linear operator, it can be thought of as an infinite dimensional matrix. Seeing the ways in which these concepts tie together can often clarify/solidify fuzzy intuition.

Answer 2

I'm a high school student, so I have no idea what a Jacobian or a manifold is, but as someone who's self-studied linear algebra and abstract algebra, I think it's pretty complex and takes a rather smart/dedicated person to pass these classes, so you're definitely smart enough to understand these concepts.

In my opinion, these fields are kind of intuitive in some ways. Honestly, I have no idea how mathematicians came up with all of this -- especially the more advanced (well, advanced to me) parts of these fields like field arithmetic or Galois theory or all of this spectral theorem and Schur decomposition/lemma stuff that I'm learning about right now. However, I've seen a lot of good online explanations, especially from looking at answers at this site, so I think I have a better intuition than some college students. If you're a college student on a strict schedule, you don't have the free time to explore maths like I do, so my guess is that it's b) and that the education system doesn't focus enough attention on comprehension and instead focuses on getting students passing grades and degrees.

Now, I know the beginning of linear algebra quite well since I've reviewed and cemented those concepts and I can definitely tell you how matrices have helped me. Maybe this and some of the links below will help you gain intuition on some things.

Matrices help us solve problems of systems of equations. For example: $$3x+2y=1$$ $$4x+5y=2$$ Using Algebra I knowledge, we can solve this using elimination. Divide the first equation by $3$, subtract the second equation by $4$ times that to get rid of $x$, divide the second equation by $\frac 7 3$ to solve for $y$ and subtract the first equation by the $\frac 2 3$ the secone equation to solve for $x$.

This is elimination, but this is the same process used for RREFing the following matrix: $$\left[\begin{matrix} 3 & 2 & 1 \\ 4 & 5 & 2 \end{matrix}\right]$$ If you RREF that matrix manually, you'll basically end up with the same row operations as we manipulated the equations. Thus, matrices and RREFing them is literally just solving systems of equations with elimination. However, it's hard to see that because there's no variables; we're just manipulating coefficients. It's even harder to see that with 3x3 and 4x4 matrices when this becomes increasingly more time-lengthy and when some people would rather use substitution or guess and check for such systems. However, with this method of matrix and RREFing, we have an algorithm that makes it much easier to do this without thinking or with a computer. By using this repetitive, boring algorithm, I bet mathematicians were able to solve systems a lot faster since they didn't need to keep track of variables and they didn't need to choose between elimination or substitution. They could just do the algorithm out and there's no thought involved. By using this kind of repetitive/familiar format of matrices, it makes solving systems of equations faster. RREFing basically solves the problem of linear systems, so using this rote method matrices and RREFing allows mathematicians to solve linear systems quickly and focus on more complex, interesting problems.

Now, hopefully, this explanation helped you understand matrices, but there are some more advanced concepts that I think other explanations can help you with best:

• Least squares approximation $\rightarrow$ This video taught me why in the world we need things like "left nullspace" and "orthogonal complements." These concepts fit into real world problems like least squares approximations and it's pretty cool how abstract concepts in linear algebra can come together in something useful like this.
• Eigenbases $\rightarrow$ This taught me how eigenbases can speed computations up and thus how eigenvectors can be useful if someone needs to apply a certain linear transformation repeatedly.
• Abstract Algebra: Theory and Applications $\rightarrow$ This online textbook shows a lot of the applications behind abstract algebra and gives us a motivation for doing it. It has a lot of practice problems and these proof problems took me hours and hours to solve, but I have a much better intuition about this field because of it.

In short, I've learned my intuition about mathematics because of good online explanations and a lot of practice problems. Explore mathematics, listen to explanations online, ask questions to your professors and possibly this forum and you will gain a better mathematical intuition about mathematical concepts that seem complex to you now, but down the road, will make much more sense. One year ago, all of the concepts I've explained above (except maybe matrices) made no sense to me, but now, I've gained a much better understanding of these concepts, even though I know I have a long way to go and I think that's probably the place you are in. In a few years, the concepts that seem bizarre and odd to us now will hopefully make a lot more sense to us then, at which point we might be struggling with even more complicated mathematical concepts. In any case, good luck and I hope all of this helped!

Answer 3

The answer, as you might guess, is b). But still, work hard. It is worth it! I have only very basic intuition and only in the area I am beginning to specialize in, but every time I gain a little insight and really feel it, the fun and the rush are worth it! Do lots of problems. Ask each and every question you have, multiple times, to multiple people. Compare your ideas with others. Why did they think that way? Try to really relish ways of thinking or results you find clever. Save them in your mental back pocket - not by memorizing, but by recalling how to see them in the first place. There are a million other bullet points. I can go on and on, with many examples, as I have this talk with students I teach every semester.

Math is hard. But we can all do it if we want to and if we put in Herculean effort, especially towards improving our weak points.

Now, if you'll excuse me, I'm off to try to figure out a question about very basic number theory that I should already know how to do, yet have worked on for 14 hours over the past five days with no success. I'm sure that when I solve it, I will feel very proud, and will have gained some intuition in the process.

Answer 4

All of these topics can be understood intuitively, they just don't tend to be taught intuitively. I remember when starting linear algebra, I already knew what a vector is so it made sense to me, but I pity someone who was trying to understand what a vector is from the axioms of a vector space, which were what was being written onto the blackboard in front of us.

Certainly the topics you mention can be intuitively understood. Intuitively, a manifold is an object of lower dimensionality than the space it's in, so a piece of paper for instance is a 2d manifold in 3d space, or a string is a 1d manifold. Intuitively, if you picture a matrix as representing a transformation of space, and apply it to a unit circle/sphere, then you get an ellipse/ellipsoid. The eigenvectors are the principle axes of the ellipse/ellipsoid and the eigenvalues are the lengths of those axes. The Jacobian I know less about but basically it's the derivative, except of something involving lots of variables, so it has lots of values.

There's some cynical reasons why it's not explained intuitively including that unlike teachers in school, lecturers aren't usually there primarily to teach, and that it's a higher level course so expectations and demands are higher. But a nicer reason is that the power of maths comes from its ability to apply to many things beyond where it was first discovered. Any intuitive understanding of something is limited to a particular domain or application, and if all you're taught is that domain then you don't appreciate or understand the generality of it. Of course in practice, teaching something in a practical context first then generalising would probably be more effective.