My textbooks says that for any $m\times n$ matrix it has $Ax=b$ is consistent for every possible $b$ as long as it has a pivot on every row.
Here is what I understand by that:
MEANING OF THE QUESTION: By the rules of matrix multiplication $b$ lives in $\mathbb R^m$ so that means spanning all possible $b$ is spanning all of $\mathbb R^m$. Also the number of pivots = number of linearly independent rows/columns.
In nxn matrices: At most there are $n$ linearly independent columns/rows and $b$ lives in $\mathbb R^n$ so IT'S POSSIBLE
In $m\times n$ matrices where $m < n$: At most there can be $m$ linearly independent columns/rows and $b $ lives in $\mathbb R^m$ so IT'S POSSIBLE
In $m\times n$ matrices where $m > n$: At most there can be n linearly independent columns/rows and $b$ lives in$\mathbb R^m$, my vectors only span up to $\mathbb R^n$ which is a subset of $\mathbb R^m$ so IT'S NOT POSSIBLE.
Is my understanding correct, is there anything I am misunderstanding something? Thanks a lot in advance.