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.
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