dimC2

In mathematics, dimC2 is a shorthand term sometimes used to denote the complex dimension of the two-dimensional complex vector space C^2. More explicitly, dimC2 is the value of dim_C(C^2), the dimension of C^2 considered as a vector space over the field of complex numbers. This dimension is 2.

The space C^2 consists of ordered pairs (z1, z2) with z1 and z2 drawn from the complex numbers. A standard basis is given by e1 = (1, 0) and e2 = (0, 1). Every vector (z1, z2) in C^2 can be uniquely written as z1 e1 + z2 e2. The fact that two vectors form a basis confirms that the complex dimension of C^2 is 2.

As a real vector space, C^2 has dimension 4, since each complex coordinate z1 and z2 encodes two real degrees of freedom. The set of all complex-linear endomorphisms of C^2, End_C(C^2), is isomorphic to the matrix space M2(C) and is thus 4-dimensional as a complex vector space (and 8-dimensional when viewed over the real numbers). The general linear group GL(2, C) describes invertible complex 2×2 matrices acting on C^2.

DimC2 is independent of the choice of basis and frequently arises in contexts such as linear transformations, representation theory, and quantum mechanics (where C^2 serves as a basic two-level system). In brief, dimC2 = 2, reflecting the two degrees of freedom in a complex two-dimensional vector space.