TITLE: Why we have introduced linear algebra?
QUESTION [7 upvotes]: 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.

REPLY [4 votes]: 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).