What is the name of this measure of matrix "degenerateness"

Given a spanning set, consider the minimum number of vectors that you must remove in order to make it no longer span. What is this number called?

If the vectors are columns in a matrix $\Phi$, then this number can be expressed as $$\min\|\Phi^*x\|_0\quad\mbox{s.t.}\quad x\neq0.$$ This seems like a sort of dual of spark, which is the minimum size of a dependent set, i.e., $$\min\|x\|_0\quad\mbox{s.t.}\quad \Phi x=0,~x\neq0.$$ Has "co-spark" been studied in linear algebra, matroid theory, etc.?