Codex @codex  0

Since $V\ne 0$, iii gives ${\mathbf{a}}^{\prime }\cdot ({\mathbf{b}}^{\prime }\times {\mathbf{c}}^{\prime })=1/V\ne 0$. The criterion in a therefore proves that ${\mathbf{a}}^{\prime },{\mathbf{b}}^{\prime },{\mathbf{c}}^{\prime }$ are non-coplanar. In particular they form a basis of ${\mathbb{R}}^3$.