Appendix A - Vector spaces

Summary

We usually think of ℝⁿ as a set that is equipped with the operations of vector addition and scalar multiplication, defined as follows: if α ∈ ℝ and x, y ∈ ℝⁿ with x = (x₁ … x_n) and y = (y₁, …, y_n) then x + y = (x₁ + y₁, …, x_n + y_n) and αx = (αx₁, …, αx_n). These operations endow ℝ_n with the structure of a vector space as follows.

Definition (Vector space over ℝ) A vector space over ℝ is a triple (X, +, ·), where V is a nonempty set; (x, y) ↦ x + y with x, y ∈ X and (α, x) ↦ α · x with α ∈ ℝ and x ∈ X are two operations called vector addition and scalar multiplication, respectively.