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