The Field Axioms of ℝ Notes

3. Vector Spaces

This subsection introduces the structure within vector spaces.

Subspaces:
A subspace

W \subseteq V

$W \subseteq V$ is a subset that is itself a vector space under the same operations.

Linear Combination:
Any expression of the form

a_1v_1 + a_2v_2 + \dots + a_nv_n

$a 1 v 1 + a 2 v 2 + \dots + a n v n$ , where

a_i \in F

$a i \in F$ and

v_i \in V

$v i \in V$

Span:
The set of all linear combinations of a set of vectors. Denoted

\text{span}\{v_1, ..., v_n\}

span{v1,...,vn}

Subspaces and spans form the gateway to understanding bases and dimension, key ideas in both linear algebra and analysis.