I consulted 1 and 2 but still have questions. What follow are modified editions of Prof Strang's picture from Intro to Lin Alg, 4th Ed: 
$\Large{{1.}}$ In the given correct version, why is the nullspace $\mathbf{Ax = 0}$ drawn right of the row space? Why not to the left?
$\Large{{2.}}$ I register that $\mathbf{A^Ty = 0} \iff \mathbf{y^TA = 0^T = 0}$.
But $N(A^T) := \{\mathbf{\color{#B8860B}{y} : A^T\color{#B8860B}{y} = 0}\}$ in which the $\mathbf{\color{#B8860B}{y}}$ is situated right of $\mathbf{A^T}$. So in the given version, why is the left nullspace drawn left, and NOT right, of the column space?
$\Large{{3.}}$ Is the flipside picture a correct substitute? It depicts
dim(row space) + dim(left nullspace) $ = r + (m – r) = m = dim(\mathbb{R}^m)$
and dim(column space) + dim(nullspace) $ = r + (n – r) = n = dim(\mathbb{R}^n)$.
How and why would this be wrong?
Please omit the following concepts which succeed this question: Orthogonality, Determinants, eigenvalues and eigenvectors, and linear transformations.
Best Answer
I think the reason the diagram is drawn the way it is in the text is to underscore that the row space and the nullspace are orthogonal complements, and similarly that the column space and the left nullspace are orthogonal complements. The order of terms (e.g. y is to the right of $A^T$) is not something he is trying to illustrate would be my guess: Focus on the depiction of orthogonality between the subspaces instead. Also notice that a major part of the illustration (at least in my version) includes arrows connecting the spaces that depict what he calls the 'true action of the matrix A.' I think your "flipside" picture could be used as a complementary illustration, but it wouldn't be correct with the remaining parts of the original diagram in which the arrows map the orthogonal dissection of A into rowspace and nullspace on the left to orthogonal dissection of column space and left nullspace on the right. I realize that you wanted to omit orthogonality from the discussion but I really think that is the issue (Strang's diagram is a synthesis of sections 2.4-2.5 and is meant to encompass these ideas) so it is impossible to interpret without considering the other aspects. Here's a photo of the diagram from my 2nd edition of Strang so you can see the arrows (I'm curious if this diagram has changed in other editions of Strang's book?):