Span

Span is the set of all possible linear combinations given input vectors.

Geometric Interpretation

It may be easier to understand span geometrically.

Consider a nonzero vector in $\mathbb{R}^3$, $\vec v$. $\text{span}(\vec v) = c\vec v$ for all possible values of $c$. If we then graphed the span, it would form a line that $\vec v$ lies on.

With two nonzero vectors in $\mathbb{R}^3$, the span can either form a plane or a line if one is a scalar multiple of the other.

Span Theorem

A common question is whether vectors span the vector space. In other words, given the vectors, can you create a linear combination for every point in the space?

Consider $A\vec x = \vec b$.

The following four statements are equivalent. If satisfied, the vectors indeed span the entire vector space:

1. For each $\vec b$ in $\mathbb{R}^m$, the equation $A\vec x = \vec b$ has a solution
2. Each $\vec b$ in $\mathbb{R}^m$ is a linear combination of the columns of $A$
3. The columns of $A$ span $\mathbb{R}^m$
4. $A$ has a pivot in every row