Why do we care if a set forms a basis? I am taking an advanced linear algebra course and am once again confused by a lot of the concepts. I understand that the definition of a basis is a set of vectors that spans the vector space and is linearly independent. Can anyone provide any intuition on why this matters?
I can visualize (kind of) what it means for a set to span a vector space, and I understand (from regression analysis) why it is important that vectors are linearly independent, but I don't really understand why it matters if they are both, ie why it matters if they form a basis.
For reference I am an undergraduate interested in statistics and data analysis. I have taken courses in mathematical statistics, regression analysis and am currently enrolled in time series analysis. I am somewhat familiar with PCA. Any intuition that can be provided through any of those lenses would be much appreciated.
 A: We care because mathematicians are inherently lazy, and having basis usually enables you to check things for a very small number of cases while being able to generalize to the whole space...
Take linear transformations as an example. If you know how a linear transformation behaves for the basis vectors, you automatically know how it behaves for the whole space.
A: This matter because we need a basis to can express in a unique way all the vectors of a given vector space.
Indeed, for example, when in $\mathbb{R^3}$ we write that a vector is $\vec w=(a,b,c)$ what we mean is that
$$\vec w=a\vec v_1 + b\vec v_2 +c\vec v_3$$
where $\vec v_1$,$ \vec v_2$ and $ \vec v_3$ are the vectors of the choosen basis.
As an intuitive geometric argument you can think that in a 3D space you need a reference system (that is a basis) to can identify points i.e. vectors in the space.

A: Keep your geometric intuition at hand.
How do we introduce a co-ordinate system in, say, a plane? We decide about the origin, then we put together two axis (usually orthogonal, but not always), and then, suddenly, every point is described via two co-ordinates, and you can calculate! In effect, you've set up two vectors (say $\vec{OX}$ and $\vec{OY}$) and you are representing points (say $A$) using the decomposition of the vector $\vec{OA}=x\vec{OX}+y\vec{OY}$.
Now, why is it that for every point $A$ we have uniquely determined $x$ and $y$ co-ordinate? It is precisely because the vectors $\vec{OX}$ and $\vec{OY}$ make up a basis of the vector space of planar vectors.
Generalising this example, I think this is the bottom line:


*

*A basis lets you set up a "co-ordinate system"; say, $\bf{e}=(\bf{e_1},\bf{e_2},\ldots\bf{e_n})$ is a basis of a vector space $V$: now every vector $\bf{v}\in V$ can be represented using "co-ordinates" $(x_1,x_2,\ldots,x_n)$ such that $\bf{v}=x_1\bf{e_1}+x_2\bf{e_2}+\ldots+x_n\bf{e_n}$ (this is because $\bf{e}$ spans the whole $V$), and those co-ordinates are uniquely determined (this is because $\bf{e}$ is a linearly independent system).

*Having the co-ordinates, we can now calculate in $V$ using ordinary calculation in the base field.

A: There is an important theorem:
All Basis have the same size.
This allows one to define the dimension of a vector space as this size. 
The dimension is the fundamental invariant of a vector space.
