It is interesting, but not necessary. Choose for example the linear map , then you can easily find that and without converting into matrix form.

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.

I don't understand the exact meaning of your question.

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

,