Past exam of the mathematical methods I course of the University of Oxford / 2017 resit / 9 / a / Solution

The vectors are linearly independent over $F$ if ${\sum }_{j=1}^n{c}_j{\mathbf{v}}_j=0$ with $c_j\in F$ implies $c_j=0$ for every $j$. Otherwise there is a nontrivial linear dependence. Forming a matrix with the displayed vectors as columns givesThusis the unique dependent case. At that value the explicit relation $2{\mathbf{v}}_1-\frac{3}{2}{\mathbf{v}}_2-{\mathbf{v}}_3=0$ verifies the dependence.