Linear Dependence and Independence

A list of vectors \(v_1, \dots, v_m\) can be either linearly dependent or linearly independent. These can be distinguished by considering of these vectors.

Let \(V\) be a vector space over a field \(\mathbb{F}\). A linear combination of vectors \(v_1, \dots, v_m \) is a vector \(v\) defined by some choice of \(m\) field elements \(a_1, \dots, a_m \in \mathbb{F}\) as \( v = a_1 v_1 + \cdots a_m v_m = \sum_{i=1}^m a_i v_i. \)

A list \(v_1, \dots, v_m\) of vectors in \(V\) is linearly independent if \(\sum_{i=1}^m a_i v_i = 0 \) implies all the \(a_i =0\). A list is linearly dependent if it is not linearly independent.

So, a linearly dependent list of vectors \(v_1, \dots, v_m\) has a choice of field elements \(a_1, \dots, a_m\) not all zero such that \(\sum_{i=1}^m a_i v_i = 0 \). For a linearly independent list, no such choice exists.

We've seen how a list \(v_1, \dots, v_m\) of vectors in a vector space \(V\) can be a spanning list if every vector in \(V\) can be written as a linear combination of these vectors. Such a list could be either linearly dependent or linearly independent. A list of vectors in \(V\) which both spans \(V\) and is linearly independent is called a basis.

Let \(V\) be a vector space over a field \(\mathbb{F}\). A basis for \(V\) is a linearly independent list of vectors in \(V\) which spans \(V\).

To actually prove a list is linearly independent or linearly dependent, we often either check the definition directly or make use of the following Lemma, 2.21 in Axler.

If \(v_1, \dots, v_m\) is a linearly dependent list in \(V\), then there exists \(j \in \{1,\dots, m\} \) such that \(v_j \in \text{span}(v_1, \dots, v_{j-1})\) and removing \(v_j\) from the list doesn't change its span.