Span & Basis

Span tells you what vectors you can build; a basis builds everything with no redundancy.

The idea

The span of a collection of vectors is the set of everything they can build: all vectors obtained by scaling them and adding the results, their linear combinations. One nonzero vector in the plane spans a line, since scaling it only moves along a single direction. Two vectors in different directions span the whole plane.

A spanning set can carry redundancy. Add a third vector to two that already span the plane and the span does not grow, because the new vector was already a combination of the first two. A set that spans the space and is also linearly independent — no vector in it a combination of the others — is a basis of the space.

A basis turns the space into a coordinate system. In the figure, $v_1$ and $v_2$ form a basis of the plane, and every vector $w$ is reached by one and only one combination $w = a\,v_1 + b\,v_2$: spanning provides the combination, and independence makes it unique. Any two bases of a space have the same number of vectors, and that number is the dimension of the space.

Ways to work on it

• Walkthrough. From coordinate axes to the idea of dimension.
• Practice. Decide whether small sets of vectors really form a basis of the plane.
• Hardest. Express a target vector in a nonstandard basis and use uniqueness of coordinates.

Not sure where to start? Take the ten-question placement test.