Most of the time studying mathematics we come across various properties like associative, commutative,...etc. These properties are obvious and sometimes I feel why at all they are given in the text. One such eg. is ! one such example of the properties I'm talking about
I don't know how to study them or what to make out of them. 1. Is my study method wrong? 2. How do I understand them intuitively? 3. How do top mathematics study study them? Kindly throw some light and provide valuable insights.
The thing is that these properties are not obvious. The problem with your study method here is that you assume that $+$ in this case is the normal addition operation which on natural numbers act commutatively $5+2 = 2+5$ etc. However without assuming that axiom of a vector space we may create vector spaces which are not commutative.
For instance, take the "semi-vector space" which consist of $2\times 2$ matrices where $+$ is interpreted as matrix multiplication. As you probably know (since you are currently looking at abstract vecorspaces) about matrices, $A\cdot B\neq B\cdot A$, thus we in this "semi-vector space" do not have commutativity.
The way to study any properties which you think "this is obvious, why do they assume this?" is to try and find examples (or look up, they should be avalible in your textbook) where they are not true.