= Solution
The original <basis> matrix has columns $\mathbf e_1,\mathbf e_2,\mathbf e_3$ and <determinant> $D=3\cdot3-4\cdot4=-7$. Evaluating the defining <cross products> gives
$$
\boxed{\mathbf e_1'=\left(-\frac37,\frac47,0\right),\quad
\mathbf e_2'=\left(\frac47,-\frac37,0\right),\quad
\mathbf e_3'=(0,0,1).}
$$
For instance, $\mathbf e_2\times\mathbf e_3=(3,-4,0)$, which must be divided by the negative denominator $-7$. These vectors satisfy $\mathbf e_i'\cdot\mathbf e_j=\delta_{ij}$, the defining <reciprocal basis> relations.
Dotting the desired expansion with the reciprocal vectors yields
$$
\boxed{a_1=\mathbf a\cdot\mathbf e_1'=1,\qquad
a_2=\mathbf a\cdot\mathbf e_2'=0,\qquad
a_3=\mathbf a\cdot\mathbf e_3'=2.}
$$
Indeed $\mathbf a=\mathbf e_1+2\mathbf e_3$. The original <basis> is not orthonormal, so taking dot products with the original vectors would not give these expansion coefficients.
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