The rank is the number of linearly independent rows/cols of a matrix. Generally, we think of linear independence as a binary property. But we could imagine an alternative definition that allows for numbers in the range [0,1]. Then, we could have fractional ranks.


I'm curious if there's any use to such generalizations of rank/independence, or if anyone has even thought about it before?



Is there a generalization of linear algebra that allows fractional ranks?
von Neumann thought about this; the keyword is [continuous geometry](http://en.wikipedia.org/wiki/Continuous_geometry). 


