#Strategies for solving difficult problems

6 messages · Page 1 of 1 (latest)

minor flare
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Hello! So I'm someone who's been trying to self-learn math, and it's going pretty well, but there's a couple things that still remain difficult to me.

Currently I'm trying to do some basic first and second year undergraduate mathematics (general, so like calculus, linear algebra, etc.) are there any strategies to how to think when solving very complex problems and/or theorems that save the day? I've added a couple images to show the level of problems that I mean, it's probably not very high level but I'm not fully used to using all these new formulas, and I seem to still have some gaps, so if someone could tell me some theorems and formulas (don't worry about the proofs, i can learn them online) that might not be as well-known/covered, but help with the majority of these kinds of problems?

Mainly linear algebra and calculus, but any field of math is fine in general.

And sorry if this should've gone in help-university, it felt sort of inbetween since it wasn't that I needed help on a specific problem, but just general tips.

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neat apex
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Where do these exercises come from? Someone talented may find the solutions of the first two exercises without hints, but typically there are techniques or theorems taught in lectures that would make them straightforward. The last one may not be very easy but would require a little concentration and common sense. Just read the courses corresponding the level of the exercises.

calm coral
neat apex
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Maybe I would add a difference of mindset between high school and graduate maths: we focus more and more on proofs and less about calculations. Learning about proofs, concepts and structures help in understanding calculations and vice versa. Knowledge on vector spaces would help in the first two exercises, knowledge on arithmetics, group theory and equivalence relations would help in the last exercise. Knowing where the recipes come from and not only learn them by heart can help in solving new problems.

minor flare
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yeah, now that you mention it, i do feel like i'm focusing on trying to learn as much theory as possible and use all of it, but i don't spend as much time going deeper into proofs and explanations behind why those things work

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thanks, i'll keep that in mind from now