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.
$\{b_1,\dots,b_d\}$lets you represent each vector $v$ by a $d$-tuple of numbers, the coefficients in the expansion of $v$ in terms of the $b$'s. So you can think of $v$ as a function from bases of $V$ to $d$-tuples of numbers. Of course, the $d$-tuple assigned to any one basis determines the $d$-tuples assigned to all other bases, by some obvious formulas. That's the physics book's notion of vector: a function from bases to $d$-tuples of numbers, subject to suitable transformation rules. – Andreas Blass May 23 2012 at 19:05