Is there some deep reason as to why the null space of a complex matrix is the complex conjugate space of the orthogonal complement to the row space? I feel like there is not, but rather that this is a matter of definition... I.e. matrix multiplication is defined without some complex conjugation action- it is defined how it is. And this coincides with taking the inner product of the rows with the vector that the matrix is being applied to in the real case. But there is no reason to suppose that anything like this would hold in different spaces- in particular when we change the inner product.
But perhaps I am missing a deeper connection here? Is there some deeper reason, or intuition, behind why the null space of a complex matrix $M$ is the complex conjugate space of the orthogonal complement to the row space? If there is, I think my lack of seeing this is due to not appreciating the geometrical significance of taking the complex conjugate space of some space. I would also appreciate any comments on how these two spaces are related.
 A: It comes down to the notion of adjoint; if $Ay=0$ then for all $x$, $\langle x,Ay \rangle = 0$ and so $\langle A^* x,y \rangle = 0$ for all $x$. Thus $y$, an arbitrary element of the null space, is in the orthogonal complement of the range of the adjoint. Note that this direction is proven immediately; you just move $A$ over as its adjoint and read it off.
Going the other direction, if you have $y$ in the orthogonal complement of the range of the adjoint, then for all $x$, $\langle A^*x,y \rangle=0$, so $\langle x,(A^*)^*y \rangle = \langle x,Ay \rangle = 0$. (Strictly speaking, in order to prove this step rigorously you must show that the space is isomorphic to its double dual, and then you identify $(A^*)^*$ with $A$ through the isomorphism.) The remaining step is to show that the only way for $Ay$ to be orthogonal to everything is if $Ay=0$; a quick way to get that is to plug in $x=Ay$ and use the positive definiteness of the inner product.
This happens for the adjoint with respect to any inner product, it is just that the definitions of "orthogonal" and "adjoint" are both relative to the inner product you are examining. Unlike in the real case (where you can prove the fundamental subspaces theorem by just comparing the definition of matrix multiplication and the definition of the real Euclidean inner product), you really do need this concept in the complex case, because complex matrix multiplication doesn't have any conjugates appearing in it.
