If we have a linear transformation $T$ represented with a matrix $A$, the first step of finding the kernel of $T$ is to construct an augmented matrix with $A$ on top and $I$ on the bottom. If we perform some column operations on the augmented matrix to get some of the columns on top to be zero, the corresponding columns in the bottom will be in the kernel of $T$. This makes sense because the column operations correspond to coordinate transformations on the domain of the matrix, so if we transform the domain so that the vectors in the image become 0, performing the same transformations on the standard basis of the space (the identity matrix) will yield elements of the kernel.
Hopefully I'm understanding that correctly. It would make sense then that the basis of the kernel is all the bottom columns corresponding to the zero columns once we've maximized the number of zero columns in the upper matrix.
So my question is, how do we know that column eschelon form has the maximal number of zero columns of any matrix that can be reached from elementary column operations? Hopefully this isn't terribly obvious.