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?

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    $\begingroup$ Dimensions of objects in monoidal categories can be arbitrary elements of the base ring, and sometimes this freedom is used: mathoverflow.net/questions/16668/… $\endgroup$ – darij grinberg Jan 31 '14 at 1:32
  • $\begingroup$ Hello Mike -- You ask first for every number in [0,1] but then for fractional ranks. Wikipedia let's a fractional part be real, but the word made me think of fractions. Are you interested in situations in linear algebra where the generalized ranks are rational? $\endgroup$ – Terry Loring Jan 31 '14 at 1:50
  • $\begingroup$ Sorry, the number between [0,1] would be an alternative definition for two rows being linearly independent. Then the rank could take on any real range from [1:n], where n is the number of rows/col of the matrix. That's just one idea I had about how ranks might be generalized. You could potentially have others, e.g. by allowing fractionally many columns in the matrix, but I this seems to make even less sense intuitively. $\endgroup$ – Mike Izbicki Jan 31 '14 at 2:26

von Neumann thought about this; the keyword is continuous geometry.

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