Real Vector Bases

Definition

A set of vectors $\set{v_1, \dots , v_k}$ is a basis for a subspace $S \subset \R ^n$ if

\[ S = \span\set{v_1, \dots , v_k} \quad \text{ and } \quad \set{v_1, \dots , v_k} \text{ is independent. } \]

This definition captures two competing properties. The first is that the set is large, in the sense that any vector in $S$ can be represented as a linear combination of vectors in $\set{v_1, \dots , v_k}$. Simultaneously, the set is small, in the sense that no vector in the set is a linear combination of the others. In other words, there is no extra vector in the set.

Linear independence is equivalent to uniqueness of representation of the vectors representable as a linear combination of $v_1, \dots , v_k$. In other words, $\set{v_1, \dots , v_k}$ is a basis for $S$ if each vector $x \in S$ can be uniquely expressed as

\[ x = \alpha _1 v_1 + \cdots + \alpha _k v_k. \]