On Nov 4, 2:22 pm, Rob <tadej.sla...@gmail.com> wrote:
> Hi all,
> I'm trying to understand what was the motivation behind "creating" > vector spaces. I know what the definition is but I find it difficult > to understand why the definition is constructed like that, what is > their "purpose" and why are they such a important part of linear > algebra.
Well, by *definition*, linear algebra is the study of vector spaces, so of course vector spaces are "such an important part" of linear algebra. What you are asking is kind of like asking why animals are such an important part of zoology, or why historical events are such an important part of the study of history... So presumably what you really mean is "why are vector spaces such an important thing?", and why we even dedicate an entire area of mathematics to study them, so much so that we give it a special name.
Part of the answer is that linear maps are very nice, and show up a *lot*. They show up all over the place. And there are many problems that are *really* hard (especially in physics, trying to model physical phenomena), but for which one can find a reasonably good approximate answer by "linearizing", that is, by pretending that the answer is a linear function. So, linear functions are pretty much the bee's knees of functions: they show up a lot, they are cool, and they are very, very useful.
So what we kind of want is to study linear maps. Vector spaces turn out to have *just* enough 'structure' to them so that you can talk about linear maps, and say useful things about them. That is, they provide you with the scaffolding necessary to be able to talk about linear maps. Now, when you build scaffolding, you are really trying to find just the right balance between not building too much and not building too little. If your scaffolding is too flimsy (too vague, too general), then it won't support you and you won't be able to build (you won't be able to say terribly much by way of useful things). On the other hand, if it is too strong, then it takes too much effort to build (you don't want a scaffolding that is just as hard to build as the building you are trying to build). The definition of vector spaces evolved through some years until it found just that perfect balance: enough structure so you can say useful stuff, but not so much that it is too specific; by keeping it as general as possible, it becomes applicable to many things. This because a desirable thing when people began to notice that many of the same arguments that were used for specific instances (linear functions of real numbers) were "essentially the same" as those used in others (linear functions on the plane; linear functions between polynomials, etc). They are the result of a process of 'abstraction', whereby people boiled away all the stuff that was extra and not needed, and kept just the essence needed to make sure all the important and interesting arguments could still be made, and the desirable conclusions reached.
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