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 / b / Solution

The columns are linearly independent. A linear combination of them with coefficients $x_1,x_2,x_3$ is exactly $Ax$. Part (a) shows that this vanishes only when all three coefficients vanish, which is the definition of column independence. Hence $A$ has column rank three.