Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Digital systems, linear algebra, introduction to formal proof / 4 / d

If $Ax=b$ has a solution, the multiplication identity from part (c) expresses $b$ as a linear combination of the columns, so $b\in \mathrm{Col}(A)$.

Conversely, if $b$ lies in that column space, its definition supplies scalars $x_1,x_2,x_3$ with $b=x_1{a}_1+x_2{a}_2+x_3{a}_3$. Put those scalars into the column vector $x$. Then $Ax=b$. This proves both directions of the claimed equivalence.