The word span can be used as a noun or a verb. Here's the formal definition.

The span of a finite, non-empty set of vectors {u1, u2, ... , un} is the set of all possible linear combinations of those vectors, i.e. the set of all vectors of the form

c1u1 + c2u2 + ... + cnun

for some scalars c1, c2, ..., cn.

Alternately, if all vectors in a set * S* of vectors are linear combinations
of a set of vectors

{u1, u2, ..., un}, then u1, u2, ..., un are said to span

**Example**. A linear combination of a single vector
is just a scalar multiple of that vector. The span of a single non-zero vector
is thus the set of all vectors parallel to that vector.

Drag the slider to see some of the vectors in the span of the red vector.

0 = 0u + 0v + 0w.

What is the span of the set containing just the
zero vector?

Vector Spans | |||

Introduction | Definition of a span | The span of two vectors | The span of three or more vectors |