Solution
= Solution
\b[Yes, over $\mathbb C$.] The column matrix is $\operatorname{diag}(1,1+i)$, with nonzero <determinant> $1+i$. Every $(z_1,z_2)\in\mathbb C^2$ has the unique expansion
$$
(z_1,z_2)=z_1(1,0)+\frac{z_2}{1+i}(0,1+i).
$$
The scalar field matters: the same space regarded as a real <vector space> has dimension four.