In physics book there are some wired definition of vector in terms of how the quantity behave under rotation of axes[Kundu P26]. Are they trying to define vector space in term of the results of special kind of map or some thing like that? or we are talking about different things? Thanks.
Kundu, Pijush K., and Ira M. Cohen. Fluid Mechanics, Second Edition. 2nd ed. Academic Press, 2001.