Definition of R^n (or C^n)

**Definition:** \(\mathbb{R}^n\) is the set of all ordered \(n\)-tuples of real numbers:

\[\mathbb{R}^n = \{(x_1, x_2, \ldots, x_n) \mid x_j \in \mathbb{R} \text{ for } j = 1, 2, \ldots, n\}\]

Similarly, \(\mathbb{C}^n\) is the set of all ordered \(n\)-tuples of complex numbers:

\[\mathbb{C}^n = \{(z_1, z_2, \ldots, z_n) \mid z_j \in \mathbb{C} \text{ for } j = 1, 2, \ldots, n\}\]

**General notation:** For any field \(\mathbb{F}\), we write:

\[\mathbb{F}^n = \{(x_1, x_2, \ldots, x_n) \mid x_j \in \mathbb{F}\}\]

**Examples:**
- \(\mathbb{R}^2\) — the Euclidean plane; e.g., \((3, -1)\)
- \(\mathbb{R}^3\) — 3D space; e.g., \((1, 0, -2)\)
- \(\mathbb{C}^2\) — pairs of complex numbers; e.g., \((1+i, 2-i)\)

**Structure:** \(\mathbb{F}^n\) is a vector space over \(\mathbb{F}\) under component-wise addition and scalar multiplication.Copy