R^n is a vector space.

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).