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:
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}).
**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 \(\mathbb{Q}\).