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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?2012-07-31
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    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.2012-07-31
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    Solutions to linear differential equations?2012-07-31
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    @QiaochuYuan if they study linear algebra I doubt they know what is a differential equation2012-07-31
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    @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.2012-07-31
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    @QiaochuYuan - I did not know that, in Israel linear algebra is a first semester course for engineers as well as math students. differential equations is at least secdond semester for engineers and at my math faclty it's a 4-th semester course.2012-07-31
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    In Italy (finite dimensional) Linear Algebra is part of the 1st year course in Geometry and it is introduced mainly with the motivations of "formalize" linear systems solutions techniques and classifying conics and quadrics. First year students see some differential equations (I would say just linear equations with costant coefficients) but I'm not sure that the words "vector space" are ever uttered in a 1st year Analysis course.2012-07-31
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    At my university (in Poland), linear algebra is taught in the first (mostly 2- and 3-dimensional over real and complex numbers) and second (abstract) semesters, while differential equations are only taught in the fourth (they have 2 semesters of analysis as a requirement). That seems reasonable enough, considering how linear algebra and multidimensional analysis are some of the very basic tools in that field...2012-07-31
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    Related: [What are some applications of elementary linear algebra outside of math?](http://math.stackexchange.com/questions/1072459/what-are-some-applications-of-elementary-linear-algebra-outside-of-math). Maybe some of the other posts tagged [linear-algebra+applications](http://math.stackexchange.com/questions/tagged/linear-algebra+applications) might be interesting for you.2016-02-10

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