I was working out the definitions of matrix rank while trying to understand the Manifold Mixup paper & found myself with the below quandry:
(A) For an 5x3 matrix , the row rank must equal the column rank. Which appeared to mean the maximum value the rank can take is 3, because there are only 3 columns so the number of basis vectors of the column space could presumably not be greater. (B) Alternatively, according my understanding of this Wikipedia article, rank can also be defined as the dimensions of its image if a matrix is a linear transformation. (C) If you use a 5x3 matrix A to transform a 3x1 vector v, you get a 5x1 vector Av. The dimensions of the image is 5x1, so from (B) the rank is 5, but from (A) the rank can at most be 3 giving an apparent contradiction.
Could someone please tell me where I am getting this wrong? I suspect the issue is coming from a confusion of the definitions of rank? But I can't quite pinpoint it.
I would greatly appreciate the assistance.