Let $A \in {\Bbb R}^{n \times m}$. If the linear system $Ax = b$ has a solution for every $b$, then is $m \geq n$?
I can't seem to find an explanation for this question. This is the only information that is given. If I am not mistaken, for this system to have a solution for every vector b, the columns of A must span the entire n-dimensional space. How does this translate into the rank of the matrix and its size?
As we know we can consider a matrix as a linear transformation. Now your assumption implies that as a linear transformation, $A$ is surjective. So $R(A)=dim(\mathrm{Im}(A))=\text{number of rows}=n$. Note that $R(A)\le min\{ m,n \}$, hence $n\le min\{ m,n \}$ , which implies that $n\le m$.