I first introduce the vector along the lines 'something with magnitude and direction'. Later on the definition of a vector becomes generic - 'an element of a vector space'.
Euclidean spaces (n=2 and n=3) are something we can all visualize. However when describing other vector spaces such as set of polynomials or set of continuous functions all this stuff becomes abstract and many finding this severely boring. Are there other vector spaces which students will find attractive? Are there any real life examples of vector spaces which would be a good hook?
Thanks in advance for any replies.