# 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.

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Which kind of basis vectors for which kind of students? While waiting for answers, why not working on your accept rate? Might also motivate some answer(er)s... –  draks ... Jul 16 '12 at 9:29
I think of them as the "skeleton" of a vector space. Often, when we have some sort of "system" of things, we break it down into "components". For example, molecular structure is often phrased in terms of the elemental composition. In a vector space, we're "adding" things and "scaling" things. So it's nice to have a "simple set" of things to add and scale, some "basic ingredients" in the recipe for a vector, sort of like how you can create any picture you want on an etch-a-sketch, with just 2 controls (manual dexterity permitting). –  David Wheeler Jul 16 '12 at 11:05
If they're already used to writing vectors as, say, $(3,5,4)$, just ask them: $3, 5,$ and $4$ of what? –  Robert Mastragostino Jul 16 '12 at 11:51
I'm probably in no place to say this and sorry if you consider this to be offensive, but maybe it would help improving your question (and thereby the answers) if you would clarify what your background is. The way you write your questions suggests that you are teaching a class on LA, while the questions themselves indicate that you are just learning the basics. I for my part find this very puzzling. –  Simon Markett Jul 16 '12 at 12:11
I would just draw them on a blackboard. I really don't see a problem with the concept "basis", it is so natural compared to other stuff, like determinants or matrices. –  Yrogirg Jul 16 '12 at 13:08

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.

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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.

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+1 for "efficient" spanning sets. Funny though, signal processing these days would beg to differ. Compressed sensing is all about using redundant spanning sets, which gives "efficient" sparse coordinates. Of course, that's a different meaning for "efficient". –  user2468 Jul 16 '12 at 14:33