How to introduce basis vectors What is attractive way to introduce basis vectors? I am looking for a hook that students will find motivating. It needs to have an impact. 
I have normally introduced it by just stating independent vectors that span the space. 
 A: You can think of a (finite) basis as introducing some kind of coordinate system on the vector space.
This agrees with our usual Cartesian coordinate system on $\mathbb{R}^n$. Each point in this space can be written uniquely in coordinates as $(a_1,a_2,...,a_n)$ but this is just the same as the vector $a_1e_1+a_2e_2+...+a_ne_n$ for the standard basis $e_i = (0,0,...,1,...,0)$ of $\mathbb{R}^n$.
Another way I like to think of bases are as the most efficient spanning sets. There are many sets of vectors that can span a space. However in these sets some of the vectors might be redundant in spanning the space (because they can be "made" from the other vectors in the set). A basis has no redundant vectors. This is exactly what is captured by demanding linear independence.
What is also a consequence of the definition of basis is that the expression of an arbitrary vector in terms of the basis vectors is unique. The span condition makes such an expression possible and the linear independence tells you that if it is possible then it is unique.
A: I believe that you want to simplify the concept. They are the vectors with the property that any other vector in the space can be constructed from them. they are just like the basis of a building.
Or, more formally,
" In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space " 
A: Well, when introducing basis vectors, you definitely have already introduced vector addition and multiplication by scalar as well as the concept of linear combination. So the motivation is: We want a simple way to write down our vectors. We don't want to have to draw the arrow each time in order to identify our vector. So how can we do that? Well, we already know that if we have some vectors, we can generate more vectors by linear combination. So we don't need to name all the vectors, we need just enough vectors to build all other vectors from them through linear combination. Then once we agreed on such a set, we only need to give the coefficients of the linear combination to tell which vector we mean. Minimal here means that if you remove even a single vector from that set, you'd no longer be able to get all vectors. Such a minimal set of vectors is called basis.
It is obvious that such a minimal set is independent because if it weren't, one could eliminate one of the vectors by just replacing it by the corresponding linear combination of the other vectors.
