The complex structure of a point x = x_1, x_2 in the plane is defined by the linear map J:R^2->R^2 J(x_1, x_2) = (-x_2, x_1), and corresponds to a counterclockwise rotation by π/2. This map satisfies J^2 | = | -I (J x)·(J y) | = | x·y (J x)·x | = | 0, where I is the identity map. More generally, if V is a two-dimensional vector space, a linear map J:V->V such that J^2 = - I is called a complex structure on V. If V = R^2, this collapses to the previous definition.
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