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.