The four fundamental subspaces in linear algebra, as discussed by Gilbert Strang , are the kernel, image, dual space kernel, and dual space image (nullspace, column space, left nullspace, row space). He calls the relationship between these "the fundamental theorem of linear algebra".
Of course the kernel and image are needed for the first isomorphism theorem, which is fundamental to understanding what a homomorphism is.
But why are the dual space image and kernel important and "fundamental"?
A similar question was asked  with one answer stating
In short, these four spaces (really just two spaces, with a left and a right version of the pair) carry all the information about the image and kernel of the linear transformation that A is affecting, whether you are using it on the right or on the left.
but this entirely avoids the issue of why the image and kernel of the adjoint of a transformation are important to understanding the transformation.
Strang's paper  seems to suggest that these subspaces may provide intuition for the singular value decomposition, but I haven't studied the singular value decomposition (SVD) yet. I'm also not sure if the SVD is fundamental enough to justify classifying the dual space kernel and image as "fundamental".
edit: Thanks for the replies :)