Past exam of the mathematical methods I course of the University of Oxford / 2013 / 5 / Solution

The original basis matrix has columns ${\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3$ and determinant $D=3\cdot 3-4\cdot 4=-7$. Evaluating the defining cross products givesFor instance, ${\mathbf{e}}_2\times {\mathbf{e}}_3=(3,-4,0)$, which must be divided by the negative denominator $-7$. These vectors satisfy ${\mathbf{e}}_i^{\prime }\cdot {\mathbf{e}}_j={\delta }_{ij}$, the defining reciprocal basis relations.

Dotting the desired expansion with the reciprocal vectors yieldsIndeed $\mathbf{a}={\mathbf{e}}_1+2{\mathbf{e}}_3$. The original basis is not orthonormal, so taking dot products with the original vectors would not give these expansion coefficients.