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.
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.
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.