This might seem a silly question, but I was wondering why mathematicians came out with more than 3 dimensions when studying vector spaces, matrices, etc. I cannot visualise more than 3 dimensions, so I am not seeing the goal of having more than 3 dimensions.

Except from the fact that the general rules of linear algebra can be applied mathematically to vectors spaces of higher dimensions, what is the practical purpose of having more than 3 dimensions?

Linear algebra is also the study of system of linear equations that can have more than 3 unknowns, is this maybe related?

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    $\begingroup$ In computer science, there are many applications where objects are represented using high dimensional vectors. For example, image/document retrieval. $\endgroup$ – PSPACEhard Jun 2 '15 at 12:06
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    $\begingroup$ I nice motivation, even historically, can be the method of least squares : en.wikipedia.org/wiki/Least_squares . It is nice to think the method as the projection of a vector in a higher dimensional space to a 2-dimensional subspace (generated by the sample) along its orthogonal complement. If you like this example perhaps you would find interesting the following book: amazon.com/Geometry-Multivariate-Statistics-Thomas-Wickens/dp/… $\endgroup$ – Holonomia Jun 2 '15 at 12:07
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    $\begingroup$ "Dimensions" in the sense of linear algebra aren't usually "spacial dimensions", as we're inclined to think of them. Vectors are not always "arrows" with "magnitude and direction". $\endgroup$ – Omnomnomnom Jun 2 '15 at 12:17
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    $\begingroup$ You might have a look at What are some applications of elementary linear algebra outside of math? and Why study linear algebra? You will see that many things mentioned there requre more that 3 dimensions. $\endgroup$ – Martin Sleziak Jun 2 '15 at 12:58
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    $\begingroup$ I think one of the reasons that linear algebra was studied was as a consequence of studying linear equations. Then it is very natural to have for example a system of 5 linear equations with 5 unknows. $\endgroup$ – wiskundeliefhebber Jun 2 '15 at 12:58

You might want to look at applied sciences. Weather forecast seems to be a nice example which shows how a rather easy question leads to high dimensional vector spaces.

Suppose you want to predict the temperature for tomorrow. Obviously you need to take today's temperature into account so you start with a function $$f:\mathbb R\rightarrow\mathbb R,~x\mapsto f(x),$$ where $x$ is the current temperature temperature and $f(x)$ your prediction. But there is more than just the current temperature you have to consider. The humidity is important as well, so you modify your function and get $$\tilde{f}:\mathbb R^2\rightarrow\mathbb R,~(x,y)\mapsto f(x,y),$$ where $y$ is the current humidity. Now, the barometric pressure is important as well, so you modify again and get $$\hat{f}:\mathbb R^3\rightarrow\mathbb R,~(x,y,z)\mapsto f(x,y,z),$$ where $z$ is the current barometric pressure. Already this function can't be visualized, as it takes a 4-dimensional coordinate system to graph it. When you now take into account, that there are many more factors to consider (e.g. wind speed, wind direction, amount of rainfall) you easily get a domain with 5,6,7 or more dimensions.


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