span

The span of a set of vectors is the collection of every vector you can possibly build from them using linear combinations — every result of scaling and adding them in all possible ways. If linear combination is the single move, the span is the complete map of everywhere that move can take you. You will also hear it phrased as 'the vectors span the set', meaning their combinations fill it out entirely.

Geometrically the span has a satisfyingly simple shape, and it always passes through the origin (because choosing all weights zero gives the zero vector). The span of one nonzero vector is a line through the origin. The span of two vectors pointing in different directions is a whole plane through the origin. Add enough independent vectors and the span grows to fill all of R^n. If a new vector lies in the span of the ones you already have, it adds nothing — that is the seed of linear dependence.

A tiny example: the span of (1, 0) and (0, 1) is the entire xy-plane, since a*(1, 0) + b*(0, 1) = (a, b) can be any point. But the span of (1, 0) and (2, 0) is only the horizontal line through the origin — the second vector points the same way as the first, so together they reach no further than one alone.