Menny's original version of the above question included the following example, which is better placed in an answer, so that it can be voted up and down. Like all answers to any CW question, this one is community wiki. The remainder of this post, unless someone else edit's it, consists of Menny's writing, so the first-person pronoun is Menny, not Theo.
My example - the Fibonacci Sequence! I'll write it in the way I intend to present it; I hope it won't bore you and give you an idea for the type of example I'm looking for.
- start by defining it. To them to wonder what is the general term.
- Define $F_{a,b}$ to be the Fibonacci Sequence that starts with (a,b,a+b,...). Emphasize they know the first two, determines the other terms (but not explicitly! (yet))
Ask them if there exists a sequence that they "really know", i.e., they can give me the general term. (someone will come up with the zero sequence (zero vector!)).
- Tell them: if I give you the general term of, say $F_{2,3}$ can you use this information to find the general terms of other sequences? (hopefully, we will discover that we can multiply by scalars)
- emphasize this great discovery - a scalar multiple of Fib seq is another Fib seq!
- Well, assume they are given $F_{2,3}$ explicitly, can they get to any other seq. by scalar multiples?
- No? OK, So I'll give you another sequence, which one do you want? (linear dependency...)
- Get to the fact that you can also add them!!!
- Take $F_{2,4}$ Is this enough? Yes? Well how do you get to $F_{0,1}$? and to $F_{\sqrt{2},1.5}$ (solving linear equations !!!)
- Well, these $F_{2,4}$, $F_{2,3}$ must by special, if we work hard and find their general terms, we would find any general term of any given Fib. seq!!!!!
- What are their main properties? you can't get to one from the other, with both you can get to everyone (this is almost the definition of a basis...!)
- Can we find three seq. like the last too with similar properties? how would we phrase the property "you can't get to one from the other" for 3 seq?
- Well, let them show\give as an exercise\ show it yourself that this cannot be.
- Ask: any two seq with the property that you can't get to one from the other, also have the property that you can get to everything with them (using scalar multi. and addition)?
- Ask: the reverse question?
- Summarize: We've seen a vector space, the fact that one vector cannot span 2-dim space, the fact the 3 are linearly dependent, the fact that 2 lin.indep. span and vice-versa.... and (I didn't write) that the zero vector does not help to span and you can always get to it.
- BUT.....this is becoming boring! we didn't find any general term yet and we are just assuming we did. BUT we can redefine a very glorious aim: find two linearly dependent sequences with their general term!
- We only "know" two sequences from High school - let's try arithmetic progressions. Well... it doesn't work.
- Let's try geometric sequence! ...work it out... It works! with q that satisfies q^2=q+1. At last, a "real" motivation for solving a quadratic equation!
- find a "basis"
give the formula for $F_{0,1}$!
Summarize - this time dividing the board into "formal part" which will have words like vector space etc. and a part with seq and "bad definition" as above.
If you also wish to talk about eigenvectors and give a more "natural" reason for using the geometric sequence - tell them that there is another "symmetry"/operation for the seq- The Shifting map. - Well, a sequence is geometric iff if and only if it is an eigenvector. Also, you can talk about larger recurrence laws, i.e. $a_n=a_{n-1}+a_{n-2}+a_{n-3}$ and get 3-dim space...
I also found such examples in the first chapter of Newman's Analytic number theory book (which is amazing leisure-time reading!!)
