In this section we will describe a compact way to indicate the elements of an infinite set of vectors, making use of linear combinations. This will give us a convenient way to describe the elements of a set of solutions to a linear system, or the elements of the null space of a matrix, or many other sets of vectors.

Span of a Set of Vectors

In Example VFSAL we saw the solution set of a homogeneous system described as all possible linear combinations of two particular vectors. This happens to be a useful way to construct or describe infinite sets of vectors, so we encapsulate this idea in a definition.

Definition SSCV (Span of a Set of Column Vectors) Given a set of vectors $S=\{\vectorlist{u}{p}\}$, their span, $\spn{S}$, is the set of all possible linear combinations of $\vectorlist{u}{p}$. Symbolically,

\begin{align*} \spn{S}&=\setparts{\lincombo{\alpha}{u}{p}}{\alpha_i\in\complex{\null},\,1\leq i\leq p}\\ &=\setparts{\sum_{i=1}^{p}\alpha_i\vect{u}_i}{\alpha_i\in\complex{\null},\,1\leq i\leq p} \end{align*}

The span is just a set of vectors, though in all but one situation it is an infinite set. (Just when is it not infinite?) So we start with a finite collection of vectors $S$ ($p$ of them to be precise), and use this finite set to describe an infinite set of vectors, $\spn{S}$. Confusing the finite set $S$ with the infinite set $\spn{S}$ is one of the most pervasive problems in understanding introductory linear algebra. We will see this construction repeatedly, so let's work through some examples to get comfortable with it. The most obvious question about a set is if a particular item of the correct type is in the set, or not.

Example ABS: A basic span.

Example SCAA: Span of the columns of Archetype A.

Having analyzed Archetype A in Example SCAA, we will of course subject Archetype B to a similar investigation.

Example SCAB: Span of the columns of Archetype B.

Spanning Sets of Null Spaces

We saw in Example VFSAL that when a system of equations is homogeneous the solution set can be expressed in the form described by Theorem VFSLS where the vector $\vect{c}$ is the zero vector. We can essentially ignore this vector, so that the remainder of the typical expression for a solution looks like an arbitrary linear combination, where the scalars are the free variables and the vectors are $\vectorlist{u}{n-r}$. Which sounds a lot like a span. This is the substance of the next theorem.

Theorem SSNS (Spanning Sets for Null Spaces) Suppose that $A$ is an $m\times n$ matrix, and $B$ is a row-equivalent matrix in reduced row-echelon form with $r$ nonzero rows. Let $D=\{d_1,\,d_2,\,d_3,\,\ldots,\,d_r\}$ be the column indices where $B$ has leading 1's (pivot columns) and $F=\{f_1,\,f_2,\,f_3,\,\ldots,\,f_{n-r}\}$ be the set of column indices where $B$ does not have leading 1's. Construct the $n-r$ vectors $\vect{z}_j$, $1\leq j\leq n-r$ of size $n$ as \begin{equation*} \vectorentry{\vect{z}_j}{i}= \begin{cases} 1&\text{if $i\in F$, $i=f_j$}\\ 0&\text{if $i\in F$, $i\neq f_j$}\\ -\matrixentry{B}{k,f_j}&\text{if $i\in D$, $i=d_k$} \end{cases} \end{equation*} Then the null space of $A$ is given by \begin{equation*} \nsp{A}=\spn{\left\{\vectorlist{z}{n-r}\right\}} \end{equation*}

Example SSNS: Spanning set of a null space.

Example NSDS: Null space directly as a span.

Here's an example that will simultaneously exercise the span construction and Theorem SSNS, while also pointing the way to the next section.

Example SCAD: Span of the columns of Archetype D.