An indexed set {v1,…,vp} of two or more vectors, with v1=0, is linearly dependent if and only if some vj (with j>1) is a linear combination of the preceding vectors, v1,…,vj−1.
The Spanning Set Theorem
Let S={v1,…,vp} be a set in V, and let H=Span{v1,…,vp}.
a. If one of the vectors in S, say vk, is a linear combination of the remaining vectors in S, then the set formed from S by removing vk still spans H.
b. If H={0}, some subset of S is a basis for H.
Theorem
The pivot columns of a matrix A form a basis for Col A.