Let $A\in M_{m,n}(\mathbb R)$,$ x\in \mathbb R^n$, and $b\in \mathbb R^m$. Using term column space of matrix, formulate a necessary and sufficient condition:
a)System $Ax=b$ have a solution
If $b \in L(A_{\cdot 1},A_{\cdot2},\cdots , A_{\cdot n})$
b)System $Ax=b$ have unique solution
If matrix $A$ have n linear independent column
c)System $Ax=b$ have solution $\forall b \in \mathbb R^m$
If matrix $A$ have $m$ linear independent column
d)System $Ax=b$ have unique solution for every $b\in \mathbb R^m$
the same as for $c)$
I do not know what is different in this question, ok for a I know that if $rankA<n$ then it does not mean that have solution for every b, but if $rankA=n$ it have solution for every b, so that mean that have pivot in every row, but other question look the same, can you help me if I miss something.