I'm trying to get my head around the four fundamental spaces in linear algebra. I believe I understand the column space of A - "C(A)" and the left nullspace - "$N(A^T)$" which are shown in the $R^m$ section of the diagram.
My question is, does C(A) with $N(A^T)$ make up the entirety of $R^m$? Or, does something exist outside of these?
I guess this comes down to whether there are vectors in the $R^m$ space that don't solve either Ax=b or Ax=0?
The column space is the $b$ such that there exists $x$ with $Ax=b$. The transpose nullspace is the b such that $A^Tb=0$. You probably know that the column space and the (non-transposed) nullspace have dimensions adding up to the dimension, say $n$, of the domain. Similarly, the row space and the transpose nullspace have dimensions adding up to the dimension, say $m$, of the range.
So to answer your question, it would suffice to show that the row space and the column space have the same dimension (because the column space and the transpose nullspace intersect only at $0$.) And this is true! See the following for many explanations. Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix
EDIT: Since you're thinking transposes anyway, bringing in the inner product makes this more obvious. $A^Tv=0$ if and only if $\langle w,A^Tv\rangle=0$ for all $w$, if and only if $\langle Aw, v\rangle=0$, that is, the transpose's nullspace is the orthogonal complement of the column space of $A$.
