Why can't vectors be defined by magnitude and direction alone?

[hokhani]

Why we can not define a vector as a quantity which has magnitude and direction? Why we define the vectors according to behavior of its components in rotated coordinate-frames?

 

[S.G. Janssens]

Vectors are, by definition, the elements of a vector space. In my opinion, other definitions are corollaries in disguise.

 

[haushofer]

Vectors and tensors in general are mathematically defined independent of coordinate systems. This emphasizes the fact that they are geometrical objects which don't care about your preference for a certain coordinate system. If you regard a vector via its components as an array {a,b,...} which you can assign a length to, it looks like vectors cannot exist without the choice of coordinates.

From the coordinate-free definition it follows that the components (!) have a certain behaviour under coordinate transformations, like the rotations you mentioned.