Past exam of the mathematical methods I course of the University of Oxford / 2019 resit / 11 / b / iii / Solution

Expand the two linear combinations and use the Pauli matrix multiplication identity:Interchanging the vectors reverses the cross product, so their commutator is $2i\sigma (\mathbf{a}\times \mathbf{b})$. The three Pauli matrices are linearly independent; alternatively, trace against ${\sigma }_k$ to recover each coefficient. ThusFor nonzero real vectors, this means they are parallel or antiparallel, equivalently linearly dependent. The algebra also extends to complex vectors using bilinear dot and cross products, with the same linear-dependence criterion; the dot product there has no complex conjugation.