# R^n is a vector space.

Put content here**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:
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- **Addition:** \((x_1, \ldots, x_n) + (y_1, \ldots, y_n) = (x_1+y_1, \ldots, x_n+y_n)\)
- **Scalar multiplication:** \(c(x_1, \ldots, x_n) = (cx_1, \ldots, cx_n)\)
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**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.
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\(\mathbb{R}^n\) is the prototypical \(n\)-dimensional real vector space. Every real vector space of dimension \(n\) is isomorphic to \(\mathbb{R}^n\).

# Parents

* Examples of vector spaces