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 S.

Example. Every vector in 2-space is has the form ai + bj, i.e. every vector in 2-space is a linear combination of i and j. No vectors outside 2-space have this form, so the span of the set {i, j} is all of 2-space, i.e. i and j span 2-space.

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.

Example. The zero vector is always in the span of any non-empty set of vectors. It's in the span of a set of vectors {u, v, w}, for example, since

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