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I'm doing an introductory talk on linear algebra with the following aim: I want to give the student students a concrete example through which they will be able to see how the many notions arise "naturally". Notions such as vector spaces, the zero vector, span, linear dependency and independency, basis, dimension, "good" bases, solving linear equationequations, and even linear maps and eigenvectors.

So, without further ado, I'll tell you what I have in mind, and the aim of this question is to ask you which example/application would you give (not only in linear algebra, but in any other basic topic ).

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 the they know the first two, determines the other terms (but not explicitly! (yet))
• Ask them if there exist 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'll 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)?
• Summarize: We've seen a vector space, the fact the 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 to 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 satisfy 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 to 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 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 should stop writing...

The aim of this post is to find some more "concrete","real" and "natural" examples in this spirit that can interest everyone who loves what we do (and give him them motivation to learn new definitions and formalizms)formalisms). I also found such example examples in the first chapter of Newman's Analytic number theory book (which is amazing leisure-time reading!!)

SO if you have some ideas - please post them! Thanks, Menny (I've made this comm wiki - please tell me if I'm mistaken to do so) (by the way - some interesting examples appear in http://mathoverflow.net/questions/17006/linear-algebra-proofs-in-combinatorics)

Post Made Community Wiki by Menny
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