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currently I am studying Vector spaces and sub spaces. I enjoyed working with matrices and using the Gaussian-Jordon elimination and I also had no problems with cofactor expansion and determinants in general. But for some reason I lost track when it came to vectors. I understand the geometrical representation of $\mathbb R^2$ and $\mathbb R^3$ and how to solve for angles and areas of a parallelogram.

I have a hard time thinking abstractly and I think that this is currently the problem why I don't grasp vector spaces. Do you have any advice on studying this material. I know I am not brilliant in math yet, but I want to study it and take more advanced topics because I see the beauty in math and how it applies to the real world. When I work through the proofs I am unable to see the turning point or the "a-ha" effect. The proofs are not in numbers so I can't even check my results if I am doing it right. Is there actually a method to train abstract thinking? I really appreciate any advice on this matter even though it is not the usually question asked here.

Thank you for your time reading this and your effort in possible answers.

-Daniel

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    So I guess this isn't really a linear algebra question. There are many resources and books in techniques of mathematical proof, and many math departments at universities offer a course dedicated to teaching students the skill. I believe "How to Prove It" by Velleman is a well-regarded book on the topic.2012-12-12
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    The trick, especially when you're first starting out, is to really think about the material for hours. There is no short-cut. Just sit and ponder, play with it, play with examples, etc. Eventually, the nature of these abstract spaces will start to become familiar.2012-12-12
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    if that helps, I'm also an engineer learning algebra, and i feel like you can stick with the concrete ever so useful stuff: matrices, linear systems, determinants, decompositions... don't worry about that abstract vector space stuff. it's a purist generalization and you can use algebra perfectly well without it2016-01-03

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