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In Halmos book "Finite Dimensional Vector Spaces", I read

One might be tempted to say that from now on it would be silly to try to preserve an appearance of generality by talking of the general n-dimensional vector space, since we know that, from the point of view of studying linear problems, isomorphic vector spaces are indistinguishable, and, consequently, we might as well always study R^n. There is one catch. The most important properties of vectors and vector spaces are the ones that are independent of coordinate systems, or, in other words, the ones that are invariant under isomorphisms. The correspondence between V and R^n was, however, established by choosing a coordinate system.

My question is why is the most important properties of vector spaces the ones that are independant of coordinate system? Also can someone also give examples of properties that are independant of coordinate system.

Bob Ueland
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Think first about geometry. The theorems in Euclid describe the properties lines and circles and the geometric figures you can build from them. Those follow from Euclid's axioms. You can prove them by introducing coordinates and thinking of a point in the plane as a pair of numbers, but that representation is not what's important about them.

The things that matter about an abstract vector space are its dimension, how its subspaces fit together, how transformations there behave. Those properties follow from the vector space axioms.

That said, sometimes you do want a particular coordinate system that's tailored to a particular problem. For example, if you want to study a translation of the plane it's convenient to choose the $x$-axis to be a line in the direction you are translating.

Ethan Bolker
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