The span of a set of vectors is everywhere you can reach with their linear combinations — every point of the form a\,\vec{u} + b\,\vec{v} + \dots, as the weights run over all numbers. It answers the question: "given these building-block vectors, what can I build?"

One non-zero vector spans a line (all its scalar multiples). Add a second vector pointing a different way and together they span the whole plane — you can reach any point at all. But if the second vector lies along the same line as the first, it adds nothing new: the span is still just that line.

The lattice of reachable points

The dots below mark a grid of combinations a\,\vec{u} + b\,\vec{v}. Rotate \vec{v} with the slider. While it points a different way from \vec{u}, the dots spread across the plane — their span is all of 2D space. Line \vec{v} up with \vec{u} and the whole grid collapses onto a single line.

Spanning space

We say a set of vectors spans the plane if their span is the entire plane. Two vectors that point different ways do it; so do three, or four — but two is the fewest you ever need in 2D. That "fewest needed to span" idea is precisely what basis and dimension will make exact. First, though, we need to pin down what makes a vector "add something new."