True or False?
If the homogeneous system $Ax = 0$ has only the trivial solution, then the corresponding non-homogeneous Ax = b is consistent for each b.
My answer is: False. Since all homogeneous systems are consistent, that $Ax = 0$ has only the trivial solution only says that the Matrix $A$ has the form $m\times n$ where $m\ge n.$ However, it is not sufficient to say if $Ax = b$ is consistent for each $b$, since a counter-example would be:
$$ \begin{eqnarray} 5x &+& 3y &=& 1\\ 5x &+& 2y &=& 1\\ 5x &+& y &=& 1 \end{eqnarray} $$ True or False?