Last semester I taught a linear algebra class that is intended to introduce young students (at a sophmore-junior level) to "abstract mathematics". It seems that a major conceptual hurdle for many of the students is understanding the definition of a vector space. More specifically, a vector space is some set of things to which we can perform the operations of addition and scalar multiplication. Despite an enormous amount of effort on my part, many of the students insisted that it makes sense to do things like "take the real/imaginary part" of a vector or look at the components of a vector.

What strategies have you found useful for getting students to understand this type of definition?

I made this community wiki -- please edit it if the question seems badly phrased.