$Ax=b$ doesn't have a solution, $Ax=0$ has infinite number of solutions?

Question

$A$ is a real matrix of size $m \times n$. If $Ax=b$ has no solution then $Ax=0$ has an infinite number of solutions. I've been told this claim is not true but can't find an opposite example.

Thank you for your help!

Answer 1

If $Ax=b$ has not solutions it simply means that b is not in the column space of $A$. But $A $ can be full rank column thus $Ax=0 \iff x=0$.

EG

$A=[e_1 e_2]$ and $b=e_3$

NOTE

What is true is that if $Ax=0$ has infinitely many solutions then $Ax=b$ has no solution or infinitely many solutions.

Answer 2

Hint: look for examples with $m=2, n=1$.

Answer 3

If $Ax=B$ has no solutions then the rank of $A$ is less than rank of the augmented matrix $[A|B]$. This means $A$ should have number of independent equations less than the number of variables, which should lead us to the conclusion that $A$ will have infinite number of solutions. But consider $A$ as a $2$ x $1$ matrix like $$2x = 7$$ and $$3x = 14$$ Here $A$ is [2 3]' and $B$ is [7 14]'. $[A|B]$ has rank 2 but $A$ will have rank 1. Thus $$Ax = B$$ will have no solution. But $$Ax = 0$$ will have one solution (not infinite). This should answer you.