Is it possible for a set of non spanning vectors to be independent? I was reading about linear spans on Wikipedia and they gave examples of spanning sets of vectors that were both independent and dependent. They also gave examples of non spanning sets of vectors that are dependent. My question is whether it is possible for a set of non spanning vectors to be independent when the number of vectors in the set is equal to the number of dimensions?
For example, if we have the following set in $R^3$,  {(1,2,0), (2,3,0), (5,7,0)}, then the vectors do not span $R^3$ and are not independent. Based on this example I have a feeling that it is not possible to for a set of non spanning vectors to be independent but I was looking for a more rigorous proof. Ideas?

Take any basis of a finite-dimensional space, and remove one of the vectors.  This will remain independent (subsets of independent sets of vectors are independent themselves), but it will no longer span the space.
