 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2013/5/solution

The original basis matrix has columns

e_{1}, e_{2}, e_{3}

D = 3 \cdot 3 - 4 \cdot 4 = - 7

. Evaluating the defining cross products gives

e_{1}^{'} = (- \frac{3}{7}, \frac{4}{7}, 0), e_{2}^{'} = (\frac{4}{7}, - \frac{3}{7}, 0), e_{3}^{'} = (0, 0, 1) .

For instance,

e_{2} \times e_{3} = (3, - 4, 0)

, which must be divided by the negative denominator

- 7

. These vectors satisfy

e_{i}^{'} \cdot e_{j} = δ_{ij}

, the defining reciprocal basis relations.

Dotting the desired expansion with the reciprocal vectors yields

a_{1} = a \cdot e_{1}^{'} = 1, a_{2} = a \cdot e_{2}^{'} = 0, a_{3} = a \cdot e_{3}^{'} = 2.

Indeed

a = e_{1} + 2 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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