Is the set of all underdetermined matrices A that solve Ax = b (for a given x and b) dense? Let's say I have two fixed vectors $x \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$, where $m < n$. There are an infinite number of matrices $A \in \mathbb{R}^{m \times n}$ that can solve this underdetermined system. My question is: are these solutions dense over the set of all matrices in $\mathbb{R}^{m \times n}$?
Please be gentle, I am not very mathematical.

The set of $m\times n$ matrices is a vector space, and the map $A\mapsto Ax$ is linear. Hence the set of solutions (in $A$) of $Ax=b$ is an affine subspace of the set of $m\times n$ matrices, and it is not dense, unless it is the whole space (which is the case iff $x=0$ and $b=0$).
