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Maybe I overlooked it, but I didn't see, in the previous answers, anything about a really geometric view of vectors. When introducing vector spaces, I like to use 2-dimensional vectors (arrows drawn on the blackboard, with the understanding that only length and direction matter, not the location on the board), with geometric definitions of addition and scalar multiplication. It is, of course, easy to explain that these geometric vectors are "really the same" as 2-component algebraic vectors (i.e., elements of $\mathbb R^2$), and also that the sameness depends on the choice of a coordinate system. This approach provides me with a lot of analogies for more complicated things that come up later in the course.