Just a general question; how do you find the rank and nullity of a linear transformation if you weren't to convert it into matrix form or is it necessary that you must in order to find rank/nullity?

Also say you have found the rank of a matrix (through the column space method) and say you have an answer [c1, c2, 2c1] (except written vertically as a vector of course) how do you know what number the dimension is of the image which is what the rank is supposed to show you.

Lastly how do you find the null space and thus kernel dimension (nullity) of the linear transformation in matrix form?

Thank you.