TITLE: Why are the four fundamental subspaces fundamental?
QUESTION [15 upvotes]: The four fundamental subspaces in linear algebra, as discussed by Gilbert Strang [1], 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 [2] 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 [1] 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 :)
To the men and women of the future that take the path I have trodden on, I'll leave some links that I found particularly helpful [3, 4, 5].

REPLY [2 votes]: After thinking about this for a while more, I'd like to write done some thoughts.
I'll take for granted that we want to study homomorphisms between vector spaces as deeply as possible.
Now, as is routine with algebra, we can consider the kernel and image and the first isomorphism theorem. 
In the finite dimensional case, I find there to be nothing fundamental about dual spaces in themselves. Rather, I think they should be viewed as a stepping stone to the adjoint matrix.
Now multiplication by a matrix is known to have two interpretations: A change of basis (vector stays the same, coordinates change), and a transformation (vector changes, coordinates stay the same). The left hand side of 
$$ [Tv, w'] = [v, T'w']
$$
corresponds to the first view, while the right hand side corresponds to the second because
$$ Tv = \sum_i [Tv, e_i'] e_i
$$
and
$$ v = \sum_i [v, T'e_i'] T^{-1} e_i.
$$
It is clear from the latter equation that the dual basis $e_i'$ change inversely to the bases $e_i$. This means that the coordinates in the matrix representation of $v$ change inversely to the bases $e_i$. Since the change of coordinates are so important, this insight in itself motivates the adjoint and the discussion of dual bases.
However, the main payoff of adjoints are found when we consider inner product spaces.
If we introduce an ismorphism between $V$ and $V^\ast$ we get an inner product and hence the formula for a self adjoint matrix
$$ <Tv, w> = <v, T^\ast w>
$$
which otherwise is unmotivated. 
Now that we have an inner product, we can consider the image $\mathrm{Im}\; T^\ast$ of our matrix $T^\ast:W\to V$ to be a subset of $V$ rather than $V^\ast$ and compare it to the kernel $\mathrm{Ker}\; T$ of $T:V \to W$. This gives us a method of finding the orthogonal complement to $\mathrm{Im}\; T^\ast$, which is a subset of $V$ rather than $V^\ast$. Finding orthogonal vectors is useful in optimization methods like least squares.