Span

Definition

Let $S$ be a nonempty subset of a vector space $V$.
The span of $S$, denoted $\text{span}(S)$ or $⟨S⟩$, is the set consisting of all linear combinations of the vectors in $S$.

We define $\text{span}(\varnothing )=\{0\}$ for convenience. A subset $S$ of a vector space $V$ generates (or spans) $V$ if $\text{span}(S)=V$.
In this case, we also say that the vectors of $S$ generate (or span) $V$.

Theorems

Theorem 1 *¹

The span of any subset $S$ of a vector space $V$ is a subspace of $V$ that contains $S$.

Moreover, any subspace of $V$ that contains $S$ must also contain $\text{span}(S)$.

Proof

This result is immediate if $S=\varnothing$ because $\text{span}(\varnothing )=\{0\}$, which is a subspace that contains $S$ and is contained in any subspace of $V$.

If $S\ne \varnothing$, then $S$ contains some vector $\mathbf{z}$; $0\mathbf{z}=0$ is in $\text{span}(S)$.

Let $\mathbf{x},\mathbf{y}\in \text{span}(S)$. Then there exists vectors $\mathbf{u}\_1,\cdots ,\mathbf{u}\_m,\mathbf{v}\_1,\cdots ,\mathbf{v}\_n$ in $S$ and scalars $a\_1,\cdots ,a\_m,b\_1,\cdots ,b\_n$

such that

$$\begin{array}{r}\mathbf{x}=\sum ^m\_i=1a\_i\mathbf{u}\_i\text{and}\mathbf{y}=\sum ^n\_i=1b\_i\mathbf{v}\_i.\end{array}$$

Then, for any scalar $c$,

$$c\mathbf{x}+\mathbf{y}=\sum ^m\_i=1(ca\_i)\mathbf{u}\_i+\sum ^n\_i=1b\_i\mathbf{v}\_i$$

is clearly a linear combination of the vectors in $S$. Hence $c\mathbf{x}+\mathbf{y}\in \text{span}(S)$. Thus $\text{span}(S)$ is a subspace of $V$. Furthermore, if $\mathbf{v}\in S$, then $\mathbf{v}=1\mathbf{v}\in \text{span}(S)$, so $S\subseteq \text{span}(S)$.

Now let $W$ denote any subspace of $V$ that contains $S$. If $\mathbf{w}\in \text{span}(S)$, then $\mathbf{w}$ is a linear combination of some vectors ${\mathbf{w}}_1,\cdots ,{\mathbf{w}}_k$ of $S$. Since $S\subseteq W$, ${\mathbf{w}}_i\in W$ for all $i=1,\cdots ,k$. Therefore $\mathbf{w}$ is a linear combination of vectors of $W$, hence $\mathbf{w}\in W$. Thus $\text{span}(S)\subseteq W$. \tag*{$||$}

Footnotes

1. Linear Algebra, Fifth Edition, by Stephen Friedberg, Arnold Insel, and Lawrence Spence ↩