Linear Dependence Lemma

Suppose v1,…,vmv_1,\dots,v_m is a linearly dependent list in VV. Then there exists j∈{1,2,…,m}j \in \{1,2,\dots,m\} such that the following hold:

  1. vj∈span(v1,…,vj−1)v_j \in \text{span}(v_1,\dots,v_{j-1})
  2. If the jthj^{\text{th}} term is removed from v1,…,vmv_1,\dots,v_m the span remains unchanged