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Coordinate vector spaces

Created over 8 years ago, updated 10 days ago

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).