Q:

What is the point of "seeing" a set of polynomials or functions as a vector space?

I just had a course in linear algebra. It seemed that the main purpose is to lay the foundations of vector spaces, show ways of solving systems of linear equations and in the end, classify some quadrics. But in certain points, they point several examples of what vector spaces could be but there is no development of these ideas. For example: The vector spaces of all polynomials of degree $\leq n$.
What got me confused is that I can use matrices to solve certain systems of linear equations, it seemed that I learned matrix operations and ideas in vector spaces for this end. But why is it important to treat those polynomials as vector spaces? What do I gain with it? Where does the study of them lead to?
I've tried to think about the inner product of these polynomials: $$\displaystyle \langle p_1(x),p_2(x) \rangle= \int_{0}^{1} p_1(x) p_2(x) dx$$
and at least making some analogies with what I learned with coordinate geometry, I guess It's possible to build a kind of geometry of these polynomials, for example:

I guess two polynomials are perpendicular if $\displaystyle \int_{0}^{1} p_1(x) p_2(x) dx=0$.
As $\langle p_1,p_1 \rangle=|p_1|^2$ then $|p_1|=\sqrt{\langle p_1, p_1 \rangle }=\sqrt{\displaystyle \int_{0}^{1} p_1(x) p_1(x) dx}$, then:
The angle between two polynomials should be 

$$\displaystyle \arccos \frac{\langle p_1,p_2\rangle}{|p_1||p_2|}= \arccos \frac{\displaystyle \int_{0}^{1} p_1(x) p_2(x) dx}{\sqrt{\displaystyle \int_{0}^{1} p_1(x) p_1(x) dx}\sqrt{\displaystyle \int_{0}^{1} p_2(x) p_2(x) dx}}$$
That Is: I have some basic notions for a "geometry of polynomials" but I have no clue of why I would use that for. And what is more troubling is that it seems  to be possible to extend from polynomials to a certain set of functions and hence, have a "geometry of a certain class of functions".
I have tried to see if there is any connection from distance from the roots of two polynomials to the distance of two polynomials but found nothing which I could consider useful of interesting. 

A:

They are most likely given as an example to show that certain collections of functions can be seen as vector spaces. One of the reasons this is handy, is that given a vector space, we can look for a basis, which reduces the complexity of considering "all the functions all at once" (which can be hard) to "just looking at the basis functions" (which can be a lot easier).
On a slightly different tack, it turns out that an extension ring of a field such that the field commutes with the entire ring can be turned into a vector space, which turns out to be handy in the study of fields, and polynomials in that field. So, for example, complex numbers can be seen as a vector space (of dimension 2) over the reals, and this extension is helpful in studying real polynomials.
I wouldn't read too much into "geometric meaning", except by way of analogy. So, for example, the inner product you mention can be used to define a notion of "distance" and "angle" between polynomials, but these aren't easy to visualize in infinite dimensions, and do not necessarily translate to "how close their graphs are".

