I am new to linear algebra and am trying to find the motivation behind defining it in such a way and need for defining it. To study $2$-D,$3$-D space we have geometry, so why do we need linear algebra then?

Why did they choose the exact properties which a vector space should have? I haven't found any satisfactory answer. The text books directly starts with the theory.

Can anyone explain please?

Edit: "Why we study linear algebra?" is different from my question. That question need the applications of linear algebra, I already know that and there are many source to answer that question. What I need is background of introduction to linear algebra, not only the history, also the motivation behind choosing the properties that need to be satisfied to be a vector space - why not extra why not less? Will taking extra or less condition give something which is not very useful? Please explain it.

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    $\begingroup$ Linear algebra is extremely useful and has applications in math itself, as well as computing and so on. For instance, your favourite websites like Netflix rely on linear algebra (matrices and operations with them and so on). $\endgroup$ – Dave Jul 28 '18 at 16:08
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    $\begingroup$ See the applications section here for some applications: en.wikipedia.org/wiki/Linear_algebra#Applications $\endgroup$ – Dave Jul 28 '18 at 16:09
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    $\begingroup$ I've heard that solving linear systems of ODEs provided a lot of motivation for developing linear algebra. Also, in order to generalize the idea of the derivative to a function $f:\mathbb R^n \to \mathbb R^m$, we need to introduce the idea of a linear transformation. The fundamental strategy of calculus is to approximate a nonlinear function locally using a linear transformation. By understanding the linear transformation, we can draw conclusions about $f$ itself. But understanding the linear transformation requires linear algebra. $\endgroup$ – littleO Jul 28 '18 at 16:26
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    $\begingroup$ Anything cool you will be doing later in your education will have linear algebra in it in one way or another. The geometrical aspects of linear algebra will be maybe, I don't know... 5-10% of all aspects you will learn. $\endgroup$ – mathreadler Jul 28 '18 at 17:07
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    $\begingroup$ If you want history, say so in the question itself, and say it clearly and explicitly. One should not have to read through a lot of comments to find out what distinguishes your question from the other one. You could edit further. $\endgroup$ – David K Jul 28 '18 at 18:03

Textbooks about linear algebra are trying to introduce a subject which is the result of many years of deep understanding of many branches of mathematics, in the sense that linear algebra is the natural common bed for mathematical constructions that were before unrelated. A book could never cover such a huge area in an introductory chapter. A wild and bold analogy in another field could be asking: why did people invented grammar, while we already had an informal understanding of eachother?

I guess some could trace linear algebra back to the invention of infinitesimal calculus by Leibniz and Newton. When you derive a function at some point, you get the slope of the tangent of the curve at this point: this tangent line is an approximation of the function around the point. This approximation is useful because it is easy to compute. But what if I want to do the same thing for a multivariable function? Then I have partial derivatives, but they are only approximation "in one direction" taken separately. I you want an approximation of your multivariable function "in all directions at one", you need to find a way to put all of these partial approximation together. That is what formalize linear algebra, and more precisely linear maps between linear vector spaces.

  • $\begingroup$ Thank you, it partially answered my question. $\endgroup$ – user579781 Jul 29 '18 at 14:35

In addition to other answers and comments:

  • linear algebra affords having a geometric intuition with things that are not naturally geometric. For instance, with linear algebra, you can deal with spaces of polynomials, of functions... and you can apply the same reasonings with polynomials, functions... in these spaces more or less as if you were dealing with vectors in $\Bbb R^n$. That explains the first difficulty in linear algebra: the definition of a vector space is quite abstract at first sight. But it's precisely that abstraction which explains the great power of linear algebra (and it's often the case in maths in general) because it's that abstraction which affords applying the same language and the same theorems to so different mathematical objects.

  • consequently, linear algebra is part of the common language of all mathematicians. It's quite frequent to hear, at a higher level, that something "is simply linear algebra" to say that what remains to do should be easy for every mathematician listening to this conversation.

Moreover, and simpler than the other points I mentioned: we need more than 2 or 3 dimensions! Many examples:

  • in physics when they add time as a 4th dimension and some advanced theories like string theory use at least 10 dimensions (!) to describe reality

  • in functional analysis, the goal is to study spaces of functions (e.g. spaces of continuous functions on an interval, or differentiable, or integrable, or being the solutions of a particular differential equation or PDE) which can even have infinitely many dimensions!

  • in statistics, you may need more to study than 2 or 3 variables because in real life, a same phenomenon may depend on lot more than 2 or 3 parameters (e.g. weather can depend on temperature, wind speed, pressure, precipitations, humidity...) or you may want to compare more than 2 or 3 parameters (for instance, compare unemployment rate in 20 different countries).

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    $\begingroup$ Also it might be worth mentioning that in physics, such as in classical mechanics, the number of parameters required to describe the state of a system (such as a coupled pendulum system) is often more than 3. So in this situation also we find ourselves working with $n$-tuples where $n$ is large. $\endgroup$ – littleO Jul 28 '18 at 19:35
  • $\begingroup$ Excellent answer, I would add that your example about functional analysis applies to physics because Hilbert spaces are fundamental for quantum theory. $\endgroup$ – Eben Cowley Jul 29 '18 at 3:23

This part from the Wikipedia article on linear algebra is nice:

It is mostly about starting with the study of solutions of systems of linear equations and continues with the development of matrix notation up to the axiomatic definition of vector spaces.

Another line is analytic geometry, or vector geometry, combining algebra and geometry. See for example:

Nicolas Bourbaki went a step further: he omitted geometric terms like point, line, etc. and thought with the treatment of linear algebra everything necessary was said.

Combined with calculus one arrives at vector calculus, for example, see:

One important extension going toward infinite many dimensions are:

An extension with linear inequalities leads to the economic important