Assume that the linear transformation is known as a matrix.
I am aware of algorithms (by elementary row operations etc.) which calculate the rank of a matrix and hence the dimension of the kernel of the matrix.
But I would like to know of some non-algorithmic way of doing the same (if it exists). Like if there is a closed form expression which gives the kernel dimension in terms of the matrix entries? I am not sure how to make it precise..but I would like to have a method which can be used as a step in a proof which depends on the kernel dimension. An algorithmic process to calculate the same is not generally amenable to a mathematical proof.
I would eventually be interested in knowing the difference between the kernel dimension and the image dimension of a linear transformation. Hence telling me a method to calculate just the above difference would also be very helpful.