Say you are in $\mathbb{R^3}$ and are given a set of directions that is contained in some plane.
The span is, by definition, isomorphic to $\mathbb{R^2}$ and thus the null basis is given by a single vector pointing in the orthogonal direction to the plane.
Now, say I give you the same set but I perturb one of the vectors a tiny wee bit. Now the span is technically isomorphic to $\mathbb{R^3}$ but the coefficients of points that are in the orthogonal direction of the plane are huge.
In this case I would like to pretend that the original set of points is contained in the plane even though it technically isn't.
How do I measure how degenerate the input set is? I.e. I want not just the rank but how close to orthognal the basis vectors are to each other.
In the case where the above metric is poor, how can I create a new basis that contains all the original vectors plus some more that make the span be better conditioned?