>>11022530In essence, you are studying systems of homogeneous linear equations and their solution sets in K^n, or at least that was the initial motivation.
You can see them internally, via a minimal spanning set, or externally (dually) as solution sets of linear equations. Duality allows you to go from one viewpoint to the other.
You can also think about them geometrically as linear subspaces of K^n (eg. lines, planes, etc. passing through 0) and think of basic linear algebra as studying intersections of such subspaces.
Then, you can study general finite-dimensional vector spaces, which are interesting because a number of things form finite-dimensional vector spaces. These are essentially the same objects as before, except that you might not necessarily have a natural choice of a basis which identifies your space with K^n. Once you fix a basis, you are taken back to linear systems etc.
This gives you the fundamental duality of linear algebra. You should think of all concepts in linear algebras as coming in pairs, one concrete (on K^n), and one abstract (on abstract vector spaces).
For example, given vector spaces V, W of dimension n and m with a choice of bases, you have correspondences:
column vector of size n <-> element of V
row vector of size n <-> element of V*
matrix of size m,n <-> linear map V -> W
transpose of a matrix of size m,n <-> dual linear map: W* -> V*
etc.
Usually, one of the descriptions makes things easier to understand, but there is no general rule for which one it should be. You acquire a flair for this with experience.
