Linear Independence

Let \(S = \{v_1, v_2, \ldots, v_n\}\) be a set of vectors in a vector space \(V\). If there exist scalars \(\alpha_1, \alpha_2 \ldots \alpha_n \in F\) such that not all of the \(\alpha_i\)'s are zero and \[\begin{aligned}\end{aligned}\], then \(S\) is said to be linearly dependent. If the set \(S\) is not linearly dependent, then it is said to be linearly independent. More specifically, \(S\) is a linearly independent set if \[\begin{aligned}\end{aligned}\] implies that \[\begin{aligned}\end{aligned}\] for any set of scalars \(\{ \alpha_1, \alpha_2 \ldots \alpha_n \}\).

The definition of linear dependence makes more sense if we consider the following proposition.

The following proposition is a consequence of the fact that any system of homogeneous linear equations with more unknowns than equations will have a nontrivial solution. We leave the details of the proof for the end-of-chapter exercises.

A set \(\{ e_1, e_2, \ldots, e_n \}\) of vectors in a vector space \(V\) is called a basis for \(V\) if \(\{ e_1, e_2, \ldots, e_n \}\) is a linearly independent set that spans \(V\).

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.