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.