Solution
= Solution
If $Ax=b$ has a solution, the multiplication identity from part (c) expresses $b$ as a <linear combination> of the columns, so $b\in\operatorname{Col}(A)$.
Conversely, if $b$ lies in that <column space>, its definition supplies scalars $x_1,x_2,x_3$ with $b=x_1a_1+x_2a_2+x_3a_3$. Put those scalars into the column vector $x$. Then $Ax=b$. This proves both directions of the claimed equivalence.