I'm working on a numerical problem where I have a method that returns a set of vectors, $\{v\}$. I am specifically interested in whether or not this set of vectors are truly linearly independent.
When I assemble these vectors as columns of a matrix, $A$, compute the rank of $A$, I get that the set is full rank. My suspicion is that this is not really the case, in the sense that numerical rounding errors have accumulated to produce vectors that are barely linear independent from each other.
I'm wondering if there are other notions, or quantities I could compute, that are similar to rank, but would capture the idea that a set of vectors may be very close to not spanning a vector space. For example imagine that in $\mathbb{R}^3$ I could have two vectors lying in the $xy$ plane, and a third vector that is very close to lying in the plane but has a very small non-zero $z$ component. The set spans $\mathbb{R}^3$, but in some sense barely so? Could I quantify that barely-ness somehow?
Apologies for vagueness here, but I'm asking after something that I'm not sure exists.