TITLE: Is there a generalization of linear algebra that allows fractional ranks?
QUESTION [6 upvotes]: 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?

REPLY [11 votes]: von Neumann thought about this; the keyword is continuous geometry.