Past exam of the mathematical methods I course of the University of Oxford / 2015 resit / 7 / iv / Solution

Yes, over $\mathbb{C}$. The column matrix is $\mathrm{diag}(1,1+i)$, with nonzero determinant $1+i$. Every $(z_1,z_2)\in {\mathbb{C}}^2$ has the unique expansionThe scalar field matters: the same space regarded as a real vector space has dimension four.