Theorem

An indexed set $\{ \vec{v}_1, \dots, \vec{v}_p \}$ of two or more vectors, with $\vec{v}_1 \neq \vec{0}$, is linearly dependent if and only if some $\vec{v}_j$ (with $j > 1$) is a linear combination of the preceding vectors, $\vec{v}_1, \dots, \vec{v}_{j-1}$.

The Spanning Set Theorem

Let $S = \{ \vec{v}_1, \dots, \vec{v}_p \}$ be a set in $V$, and let $H = \text{Span} \{ \vec{v}_1, \dots, \vec{v}_p \}$.

a. If one of the vectors in $S$, say $\vec{v}_k$, is a linear combination of the remaining vectors in $S$, then the set formed from $S$ by removing $\vec{v}_k$ still spans $H$.

b. If $H \neq \{ \vec{0} \}$, some subset of $S$ is a basis for $H$.

Theorem

The pivot columns of a matrix $A$ form a basis for Col $A$.