Verification: All ten axioms follow directly from the corresponding properties of real numbers. For instance, commutativity holds because (x_i + y_i = y_i + x_i) for each component.
(\mathbb{R}^n) is the prototypical (n)-dimensional real vector space. Every real vector space of dimension (n) is isomorphic to (\mathbb{R}^n).
**Theorem:** \(\mathbb{R}^n\) (the set of all ordered \(n\)-tuples of real numbers) is a vector space over \(\mathbb{R}\) with component-wise operations:
- **Addition:** \((x_1, \ldots, x_n) + (y_1, \ldots, y_n) = (x_1+y_1, \ldots, x_n+y_n)\)
- **Scalar multiplication:** \(c(x_1, \ldots, x_n) = (cx_1, \ldots, cx_n)\)
**Verification:** All ten axioms follow directly from the corresponding properties of real numbers. For instance, commutativity holds because \(x_i + y_i = y_i + x_i\) for each component.
\(\mathbb{R}^n\) is the prototypical \(n\)-dimensional real vector space. Every real vector space of dimension \(n\) is isomorphic to \(\mathbb{R}^n\).