I have nothing against starting out with motivating examples, obviously they are needed for understanding. But they should motivate the definition of a vector s

I have nothing against starting out with motivating examples, obviously they are needed for understanding. But they should motivate the definition of a vector space. Not the definition of vectors as mappings of indices. Functions are actually a great motivating example for the definition of a vector space, precisely because they are first look nothing like what student think of as a vector.