Why are the four fundamental subspaces fundamental?

Suppose you are trying to solve $Af=g$ where $A$ is a linear operator. As always, there are two big issues: existence and uniqueness. Start with a Hilbert space for the sake of discussion.

Existence: A solution of $Af=g$ exists iff $g$ is in the range of $A$. If $A$ happens to have a closed range, then the issue of whether or not $g$ is in the range is settled by looking at the kernel of $A^*$: $$ \mathcal{R}(A)=\mathcal{N}(A^*)^{\perp} $$ So, in order to obtain a solution of $Af=g$, you must impose conditions on $g$: $$ (g,k) = 0 \mbox{ for all } k \in \mathcal{N}(A^*). $$ For a lot of differential equations, you find $\mathcal{N}(A^*)$ is finite-dimensional, and so a finite number of conditions on $g$ will lead to a solution.

Uniqueness: If you have one solution of $Af=g$, then you can find all others by adding an arbitrary element of $\mathcal{N}(A)$ to $f$. That is, $$ Af_1 =g ,\; Af_2 = g \implies f_1 - f_2 \in \mathcal{N}(A). $$ Conversely, if you have a solution $f$ of $Af=g$, then $f+n$ is also a solution for all $n \in \mathcal{N}(A)$. So, uniqueness may also require adding some conditions.

This is more of an intuitive perspective, based on the vector spaces involved:

The null space of a matrix is a subspace of the same vector space that the row space is a subspace of. In this sense the row space of a matrix determines the null space, as we can define the null space of a matrix $A$ as $\{u:vu=0$ where $v^t$ is in the row space of $A\}$. This complementary relationship certainly is fundamental in characterizing a matrix.

If you are looking for applications where row space is important: suppose we have an $m \times n$ transformation $G$ and an $k \times n$ matrix $F$, and we want to know whether $F$ divides $G$ on the right, that is whether there exists $H$ such that $G=HF$. It turns out that this is true iff the row space of $G$ is a subspace of the row space of $F$.

What about column space...as with the relationship between row space and null space, the complementary space to column space is the null space of $A^*$, and the larger vector space context (which these spaces are subspaces of) in this case is different from that of the vector space containing row space and null space, in the case where a matrix is not square.

A very important application of this space which is complementary to column space, is least squares approximation for an over specified system. For a least squares solution we have $$A^*(Ax_0-y)=0,$$ that is for $x_0$ to be a least squares solution we must have $Ax_0-y$ is in the left kernel! Intuitively, to find the point $Ax_0$ where the column space of $A$ is closest to $y$ we must find a vector in the orthogonal complement of the column space of $A$ that contains $y$ and $Ax_0$ (thereby minimizing the distance).

So different contexts place emphasis on different fundamental subspaces. In most common applications there is probably just more of an emphasis on null space and column space.

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.