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C^n is a vector space.

Created over 8 years ago, updated 25 days ago

Theorem: (\mathbb{C}^n) (the set of all ordered (n)-tuples of complex numbers) is a vector space over (\mathbb{C}) with component-wise operations:

  • Addition: ((z_1, \ldots, z_n) + (w_1, \ldots, w_n) = (z_1+w_1, \ldots, z_n+w_n))
  • Scalar multiplication: (c(z_1, \ldots, z_n) = (cz_1, \ldots, cz_n)) for (c \in \mathbb{C})

Verification: All ten axioms follow from the field properties of (\mathbb{C}).

(\mathbb{C}^n) is the prototypical (n)-dimensional complex vector space. It plays a central role in quantum mechanics, signal processing, and many areas of applied mathematics.