Let's say that you calculate the basis of the kernel and it spans one vector. Then let's assume the rank of the matrix is two. Normally, you would choose the columns corresponding to the pivot points of the matrix as the basis of the image. However, could they just be two linearly independent vectors that are also linearly independent from the basis of the kernel?
I ask because by the dimension theorem, any basis of the vector space (in this case 3 dimensional) consists of the direct sum of the kernel basis and image basis of a linear transformation. Assuming we can pick two arbitrary linearly independent vectors to the kernel as the basis of the image, then there exists a basis transformation mapping that would map those two vectors to the two column vectors, so they would still be a valid basis of the image, right?