{c}

Since $V\ne0$, <1/d/iii> gives $\mathbf a'\cdot(\mathbf b'\times\mathbf c')=1/V\ne0$. The criterion in <1/a> therefore proves that \b[$\mathbf a',\mathbf b',\mathbf c'$ are non-coplanar]. In particular they form a <basis> of $\mathbb R^3$.

Solution

Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics IV / 1 / d / iv / Solution

 Solution (source code)