In my book on linear algebra (Lay), Lay writes
A system of linear equations is said to be homogeneous if it can be written in the form $A\vec{x} = \vec{0}$, where $A$ is an m x n matrix and $\vec{0}$ is the zero vector in $\mathbb{R}^m$. Such a system always has at least one solution, namely, $\vec{x}=\vec{0}$ (the zero vector in $\mathbb{R}^n$).
Seeing as both $\vec{x}$ and $\vec{b}$ are column vectors, how come they're not both in $\mathbb{R}^m$, the number of rows in the coefficient matrix $A$? I tried to look back in the book for an explanation, but could not find the reason; have I completely misunderstood the concept of $\mathbb{R}^n$ and $\mathbb{R}^m$?