There are two "formal" definitions of vectors (and tensors in general) which I've learned. The first is what I consider the "better" definition, one I learned in linear algebra. We call a set X a vector space over a field F whenever that set has properly defined operations of scalar multiplication and addition, and follows certain properties. The elements of X are then called vectors. The second definition, the one I learned as a physics major, is the one I consider the "pain in the ***" definition, that is, it is a vector is a "group of numbers" that transform a certain way (transformation law) into a new coordinate system. I've always wondered if these definitions were, in fact, equivalent and I've seen it mentioned that they are. However, what makes me doubt that they are really equivalent definitions is the emergence of "pseudovector" in the case of the second definition. These are vector-like objects which do not transform correctly upon inversion. My questions are: Consider a vector that meets the first definition; does it meet the second? Consider a vector that meets the second definition. Does it meet the first?