Let $A$ be an $N\times N$ real matrix, and let $x\neq 0$ be such that $Ax=0$.
Let me denote here with $A_i$ and $A^i$ the $i$-th row and $i$-th column of $A$, respectively.
There are two natural ways to understand geometrically what $Ax=0$ means. Thinking in terms of the raw space, it says that $x$ is a vector in the intersection of the planes whose normals are $A_i$.
On the other hand, thinking in terms of the column space of $A$, $Ax=0$ is equivalent to the columns $A^i$ being linearly dependent, and $x$ being a nonzero vector such that $\sum_i x_i A^i=0$.
While the equivalence between these two views is straightforward from an algebraic point of view, I do not understand it geometrically. In particular, I don't have any intuition about the relation between $\{A_i\}_i$ and $\{A^i\}_i$.
As an example, consider the case \begin{align} A^1 = (1,0,0), \quad A^2 = (1,1,0), \quad A^3 = (0.1,1,0). \end{align} You can see in the figure below the visualization of these three vectors (red dashed arrows), of the corresponding row vectors $A_1=(1,1,0.1)$ and $A_2=(0,1,1)$ (blue arrows), and the planes whose normals are $A_1$ and $A_2$.
The "duality" I am referring to is seen here in the fact that any point in the intersection of the two planes, if multiplied component-wise with the columns $A^i$, produces a set of three vectors summing to the origin. Thinking in terms of this figure, this seems quite magical to me.
Is there a way to understand geometrically what this "dualism" represents?