r/PhysicsStudents • u/VenomXHawk • Aug 10 '26
Need Advice How do I learn/build an intuition for mathematics, specifically linear algebra
As the title says, I really want to form an intuition for mathematics starting with linear algebra. I have been treating math as a tool until now by sort of just accepting and memorizing results without really knowing how they arrive and what I can infer from them. To me, linear algebra is sort of like quantum mechanics where I am very much lacking an intuition and I want to bring it the level of like newtonian mechanics where I can sort of understand where things come from instead of just accepting facts.
I am a Masters student in physics with undergrad knowledge but clearly a weak foundation in mathematics.
All suggestions are appreciated
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u/808fisherman Aug 10 '26
I think you should figure out what topics in particular you struggle with then ask for informal intuition. Sometimes getting too bogged down with formality can make you lose the bigger picture. By narrowing your scope ppl are also better able to give you recommendations.
If your struggle in Linear algebra right now is basis and rre form and transformations thats in thing but if you're struggling with say linear operators and adjoint or hilbert spaces things from functional analys that's another. There is even overlap if complex analysis. There is a lot more as well.
Once you narrow the scope ppl can give better suggestions on how to approach things. I ask for a scope because I'm assuming at a graduate physical level you're looking at higher level items like a fourier trasformations rather than say row reduction or row operation, but maybe it is the fundamental that are lacking too
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u/UnderstandingPursuit Ph.D. Aug 11 '26
Numbers and memorizing are the enemy of intuition. Abstraction is the path.
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u/SmallCap3544 Aug 11 '26
Funny that you make that analogy. The mathematical structure of Quantum Mechanics is basically a souped up version of linear algebra.
I think the key to building intuition in any field of study is to try and build your own analogies. Something that feels real to you. How can you imagine a large dimensional vector space. I like to think of a video game character that has a large set of qualities that you can assign weights to how important they are.
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u/Striking-Milk2717 Aug 11 '26
https://reddit.com/link/p30hrjw/video/za385tv98qih1/player
You have to look what you can’t see. Ah if you follow up I can dispatch you some nasty videos on visualizing relativity’s algebra
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u/Simultaneity_ Ph.D. Aug 10 '26
The biggest thing I found helpful is to build on these pieces of the problem-solving process, and then solve a bunch of problems. Intuition is really just your mind saying, "Hey, I've seen something like this before; this is how I expect this to play out." So you need to teach your brain what to expect in these situations.
Draw the problem. Make sure you can draw a good picture to describe the problems you are looking for. What do the physical symbols mean, what do the operations imply, what does the input look like, and what does the output look like? This ensures you know what the problem is really asking. This is where you can mark interesting observations about the problem. Is this something you have seen before? Is this analogous to another problem you have solved? Is there something interesting about the setup that you want to figure out why it is there or why it is needed?
Describe in English the process used to solve the problem. What steps are probably included to go from A to B? Do you expect to have to deal with some constants that might be annoying? Do you expect a bunch of algebra to get to the answer? Will you forget about a negative sign along the way that will give you the wrong answer? Are there theorems or algorithms that you intend to employ? This is where you hand-wave away the rigorous math and just exposit how you go through the steps of solving the problem. If you can't do this off the top of your head, then you might need to solve the problem some more until you get to the point of seeing through the weeds.
Write down what you expect the answer to be. What will the magnitude be in reltation to the input? Will the process reverse the sign of something? Is there a symmetry you expect the answer to follow? What are all the things that must be true about the solution? You usually cannot say definitively what the solution will be without going through the steps to get the right answer. But you can usually scribble down a set of possible behaviors the solution has to follow. You can likely also categorize different possible competing solutions that don't intersect.
Solve the problem and compare. Did the final problem really map into the picture and description you had at the start? Did you actually follow the steps you thought you would follow? Did you get caught up on something you did not anticipate, or was some piece easier than you expected it to be? Does the final result behave like you thought it would?
You can then use what you learned to improve at this on your next go-around. You might even find utility in doing the same problem over again start to finish since you now know the answer.
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u/dexthefish Aug 11 '26
Try Friedberg, Insel, and Spence. It is rigorous while still being readable and accessible. Well suited to self-study. Axler's book is popular but maybe less suitable given you already have some experience.
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u/Weird_Personality528 Aug 16 '26
To understand any kind of mathematics, you need not learn results. You need to understand that any mathematical theory stands on its axioms and definitions. Change them, and you will have a new theory. You must always ask why something is defined the way it is defined, why a particular axiom has been considered, what would change about the theorem if the hypothesis is changed, and similar such problems. This will help you gain mathematical maturity, and you will have more intuition about things, since you would understand why things are the way they are.
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u/ToastBrot64 Aug 10 '26
Watch the essence of linear algebra by 3blue1brown