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Students familiar with Euclidean space find the introduction of general vectors spaces pretty boring and abstract particularly when describing vector spaces such as set of polynomials or set of continuous functions. Is there a tangible way to introduce this? Are there examples which will have a real impact? I would like to introduce this in an engaging manner to introductory students. Are there any real life applications of general vector spaces?

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What is real life? – Andrea Mori Jul 31 '12 at 12:22
Well, it explains why you can add matrices/polynomials/functions and multiply them by a scalar. They behave the same way, although they are different objects, and the reason why is because they form a vector space. – M Turgeon Jul 31 '12 at 12:45
Solutions to linear differential equations? – Qiaochu Yuan Jul 31 '12 at 13:11
@QiaochuYuan if they study linear algebra I doubt they know what is a differential equation – Belgi Jul 31 '12 at 13:15
@Belgi: introducing abstract vector spaces before introducing something as fundamental as the classical harmonic oscillator or the wave equation strikes me as backwards. I don't know what the situation is like in the UK (where the OP is from) or Israel but in the US many students take differential equations before linear algebra. – Qiaochu Yuan Jul 31 '12 at 13:23

Many years ago I was having a beer with a couple of fellow math grad students at some place around Harvard Square, and we overheard some guy at the next table trying to impress a girl telling her that he was taking a Linear Algebra course which was "so difficult" having to deal with spaces of "many dimensions".

I am not sure that this Linear Algebra technique of picking up girls at a bar may be listed among "real life applications", but, if it works, sure offers an important motivation.

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Because girls don't like vector spaces with small dimension :P – Belgi Jul 31 '12 at 13:05

What's boring about polynomials and real-valued functions ?

Polynomials have a great use in science, mainly in approximations using interpolations.

Since the set of polynomials with degree smaller than $n$ is a vector space, we can take an orthonormal basis for it and easily find approximation for any real value function (depending on the inner product of course). note that the reason we can do this is that the real valued functions are also a vector space!

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Is the stock exchange real enough? OK, you'll have to abstract from the fact that you only can buy or sell complete stocks, not arbitrary fractions of stocks (although given the modern way of trading stocks, I'm not even completely sure if you really can't ;-)). But it is clear that if you e.g. put an order to sell two Microsoft stocks and buy one Apple stock, and then put another order to buy two Apple stock and five IBM stock, you ultimately sell two Microsoft stock and buy three Apple and five IBM stock. So you've got vector addition (-2,1,0,0,....) + (0,2,5,0...) = (-2,3,5,0...). This also has the advantage that there's no natural order of your basis (there's no reason why you should list Microsoft stock first, or even why you should put all stock in a linear order). Multiplication by a scalar is also clear: You buy/sell e.g. twice as many stocks.

You also get a natural dual space: The price. It is a mapping from a vector (sell/buy order) to a scalar (money to pay/earn), and it is obviously linear (if you buy one Apple and two Microsoft stocks, you pay the price of an Apple stock plus twice the price for a Microsoft stock. Stock exchange covectors are regularly listed in certain newspapers and on certain web sites.

Also it has the advantage of not having an intrinsic scalar product (you can't meaningfully ask about the product of the order "selling two Microsoft stocks" and the order "buying one Microsoft stock and selling three Apple stocks"). When coming from Euclidean vector spaces, you are inclined to take a scalar product for granted, so having a space where a scalar product simply doesn't make sense is probably a good idea.

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I don't know if this is what you are looking for, but...

The functioning of the 4G-smartphones depends on the phones ability to quickly carry out certain transformations (DFT/IDFT) in certain (for example) 1024-dimensional subspaces of the space of (periodic) functions.

Look up OFDM for more details.

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Depending on how much depth you want to introduce, I think you should mention fourier analysis. Even if they haven't taken differential equations courses before, showing that functions form a vector space is quite trivial. Once you have this, you know you can introduce the idea of a basis for this space, which lets you reliably decompose certain functions as being made up from other ones. Applications are obviously abounding. Maybe not "obviously" for your students, but take your pick to wow them, since it probably has a use in whatever they're studying.

The point is that this concept of "representing a function as a sum of others" would be difficult to justify, or to actually come up with as a concrete method. It's the same ideas as vector projection and finding a basis, but on things that don't look like vectors at all. The abstract idea of vector spaces lets you carry all these neat and powerful tools over to other problems.

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In many Mathematical problems practical or theoretical we have a Set which may be sequence of numbers,continuous Functions etc,In which addition,subtraction,multiplication and division of two number belongs to that set.This suggests the concept of vector space

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