Mathematicians are well aware of complaints like these about introductions to their subjects, by the way.
It is for a reason that this book introduces the theory of abstract vector spaces and linear transformations, rather than relying on the crutch of intuition from Euclidean space. If you want to become a serious mathematician (and this is a book for such people, not for people looking for a gentle introduction to linear algebra for the purposes of applications) at some point it is necessary to rip the bandaid of unabstracted thinking off and engage seriously with abstraction as a tool.
It is an important and powerful skill to be presented with an abstract definition, only loosely related to concrete structures you have seen before, and work with it. In mathematics this begins with linear algebra, and then with abstract algebra, real analysis and topology, and eventually more advanced subjects like differential geometry.
It's difficult to explain to someone whose exposure to serious mathematics is mostly on the periphery that being exposed forcefully to this kind of thinking is a critical step to be able to make great leaps forward in the future. Brilliant developments of mathematics like, for example, the realisation that "space" is an intrinsic concept and geometry may be done without reference to an ambient Euclidean space begin with learning this kind of abstract thinking. It is easy to take for granted the fruits of this abstraction now, after the hard work has already been put in by others to develop it, and think that the best way to learn it is to return back to the concrete and avoid the abstract.
It is for a reason that this book introduces the theory of abstract vector spaces and linear transformations, rather than relying on the crutch of intuition from Euclidean space. If you want to become a serious mathematician (and this is a book for such people, not for people looking for a gentle introduction to linear algebra for the purposes of applications) at some point it is necessary to rip the bandaid of unabstracted thinking off and engage seriously with abstraction as a tool.
It is an important and powerful skill to be presented with an abstract definition, only loosely related to concrete structures you have seen before, and work with it. In mathematics this begins with linear algebra, and then with abstract algebra, real analysis and topology, and eventually more advanced subjects like differential geometry.
It's difficult to explain to someone whose exposure to serious mathematics is mostly on the periphery that being exposed forcefully to this kind of thinking is a critical step to be able to make great leaps forward in the future. Brilliant developments of mathematics like, for example, the realisation that "space" is an intrinsic concept and geometry may be done without reference to an ambient Euclidean space begin with learning this kind of abstract thinking. It is easy to take for granted the fruits of this abstraction now, after the hard work has already been put in by others to develop it, and think that the best way to learn it is to return back to the concrete and avoid the abstract.