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.