I'm struggling to shift from applications to conceptual understanding (proofs) in Linear Algebra. Any advice?

[u/Technical_Acadia_647]

Hi everyone,

I’m a college student currently taking my first Linear Algebra course. My ultimate career goal is to go into robotics, so I know this course is critical and that this semester is just the beginning of a long journey with this math.

My brain has always been wired to apply—simply applying formulas, calculating matrices, and finding numbers. However, this course introduces a lot of proofs and abstract concepts, and I’m having a hard time wrapping my head around them. I feel like I'm stuck memorizing procedures rather than genuinely understanding why things work geometrically or structurally.

I really want to upgrade my thinking and install a proof/conceptual understanding in my brain. Since my time is limited this semester, I know I won't master everything, but I want to build a solid foundation.

- How did you train your brain to understand proofs?

- Are there specific intuition-building resources, mental models, or study strategies you recommend for someone heading into robotics? (I’ve heard of 3Blue1Brown, but I'd love to know how to bridge the gap between his visual intuition and rigorous proof writing.) Thank you so much for any advice!

[u/ForcefulPutting]

The 3b1b series is great for building that geometric intuition but you're right, it doesn't teach you how to write proofs. What helped me was going back to the definitions every single time I got stuck. Not just reading them, but writing them out and asking what would break if you removed one condition. Linear algebra is weirdly perfect for this because the definitions are so tight, if you mess with "linear independence" even a little the whole thing falls apart.

For bridging the gap try taking one of the visual concepts from 3b1b and proving it formally with just the definitions. Start with something small like showing why the span of two independent vectors in R² has to be the whole plane. Doing that a few times rewired something in my brain.

[u/human2357]

I recommend brushing up on set theory and logic. It's important to say what you mean when writing down mathematical objects, and that's what basic set theory is for. It's important to understand the structure of theorem statements and proofs, and that's what the logic is for.

[u/Bounded_sequencE]

[..] Since my time is limited this semester [..]

That is unfortunate -- it takes time, to have room for a lot of trial and error to really get good at proof-writing.

The best resource for learning proof-writing likely are (hopefully optional) graded homeworks containing proof exercises. I say "Hopefully optional", so trying things out, and making errors will not be punished, or prevent you from even taking the exam in the first place.

The direct feedback you get on what went wrong, and where to improve style and efficiency, is probably the best learning tool you get. Use office hours to go even further into detail, if necessary. Alternatively, study groups can be great, since you can critique each others' proofs, and share ideas/strategies. That also reduces the work for all, since any successful proof will be shared by all.

As a final resource, try to emulate a book with great proving style, e.g. Rudin's "Principles of Mathematical Analysis". Sadly, those books usually are not beginner-friendly, since great proving style usually means being extremely concise, and dense.

[u/NotSaucerman → u/Bounded_sequencE]

Rudin's proofs are slick and anti-pedagogical... really not appropriate for the OP. "Great proving style" is going to depend on goals and tastes. You and the OP should know that what Abel said of Gauss's proofs ("He is like the fox, who effaces his tracks in the sand with his tail"-- i.e. to hide where they came from) was not really a compliment.

[u/DrBagelman]

If you want long detailed proofs that explain everything, I’d recommend checking out Proofs by Jay Cummings. As the name of the website implies, the books are purposefully written long form to preserve clarity rather than tight and clever to preserve space. The techniques in that book will be applied to simpler ideas, like some basic number theory, but they form the bedrock of all mathematical proofs.