The Field Axioms of ℝ Notes

3.1. Examples and Non-Examples

This subsection grounds the abstract definition in concrete examples.

Examples of Vector Spaces:

• $ℝ^n$

  Rn: Standard Euclidean space (e.g.,

  $ℝ^2$

  R2,

  $ℝ^3$

  R3)

• The space of all polynomials

  $\mathbb{P}$

  P

• The set of all real-valued continuous functions on an interval

• $M_{m \times n}(ℝ)$

  Mm×n​(R): All real

  $m \times n$

  m×n matrices

Non-Examples:

• $ℕ$

  N under normal addition and scalar multiplication (no additive inverses)

• Any set where scalar multiplication is not defined over a field

Emphasize why these are or aren't vector spaces—this develops students' intuition.