Linear Combinations

Let V be a vector space and $s\subseteq V,s\ne \varnothing$. $v\in V$ is a linear combination of vectors in $S$ if $\exists u_1,\dots ,u_n\in S$ and $a_1,\dots ,a_n\in F$ such that $v=a_1{u}_1+\cdots +a_n{u}_n$

Spans

• Let $s\ne \varnothing ,s\subseteq V$. The span of $s$ (denoted $\text{span}(s)$) is the set of all linear combinations of the vectors in $s$
• $\text{span}(\varnothing )=\{0\}$
• The span of any subset $S\subseteq V$ is a subspace of $V$. Any subspace of $V$ that contains $S$ must also contain the span of $S$. (Theorem 1.5)

Generating

• Subset $S\subseteq V$ generates (or spans) $V$ if $\text{span}(S)=V$