Definition: The coordinate vector space(\mathbb{F}^n) is the set of all ordered (n)-tuples of elements from a field (\mathbb{F}), equipped with component-wise addition and scalar multiplication.
For (\mathbb{F} = \mathbb{R}), we write: [\mathbb{R}^n = {(x_1, x_2, \ldots, x_n) : x_i \in \mathbb{R}}]
Scalar multiplication:(c(x_1,\ldots,x_n) = (cx_1, \ldots, cx_n)) for (c \in \mathbb{F})
Example: In (\mathbb{R}^3), the vector ((1, 2, 3)) represents a point in three-dimensional space. Adding ((1,2,3) + (4,5,6) = (5,7,9)), and scaling (2 \cdot (1,2,3) = (2,4,6)).
Coordinate vector spaces are the prototypical finite-dimensional vector spaces; every (n)-dimensional vector space over (\mathbb{F}) is isomorphic to (\mathbb{F}^n).
**Definition:** The *coordinate vector space* \(\mathbb{F}^n\) is the set of all ordered \(n\)-tuples of elements from a field \(\mathbb{F}\), equipped with component-wise addition and scalar multiplication.
For \(\mathbb{F} = \mathbb{R}\), we write:
\[\mathbb{R}^n = \{(x_1, x_2, \ldots, x_n) : x_i \in \mathbb{R}\}\]
**Operations:**
- **Addition:** \((x_1,\ldots,x_n) + (y_1,\ldots,y_n) = (x_1+y_1, \ldots, x_n+y_n)\)
- **Scalar multiplication:** \(c(x_1,\ldots,x_n) = (cx_1, \ldots, cx_n)\) for \(c \in \mathbb{F}\)
**Example:** In \(\mathbb{R}^3\), the vector \((1, 2, 3)\) represents a point in three-dimensional space. Adding \((1,2,3) + (4,5,6) = (5,7,9)\), and scaling \(2 \cdot (1,2,3) = (2,4,6)\).
Coordinate vector spaces are the prototypical finite-dimensional vector spaces; every \(n\)-dimensional vector space over \(\mathbb{F}\) is isomorphic to \(\mathbb{F}^n\).