F^n is a vector space.

Theorem: For any field (\mathbb{F}), the set (\mathbb{F}^n) (all ordered (n)-tuples of elements from (\mathbb{F})) is a vector space over (\mathbb{F}) with component-wise operations:

• Addition: ((a_1, \ldots, a_n) + (b_1, \ldots, b_n) = (a_1+b_1, \ldots, a_n+b_n))
• Scalar multiplication: (c(a_1, \ldots, a_n) = (ca_1, \ldots, ca_n)) for (c \in \mathbb{F})

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

This generalizes (\mathbb{R}^n) and (\mathbb{C}^n) to any field, including finite fields like (\mathbb{F}_p) (integers modulo (p)) and the rational numbers (\mathbb{Q}).