Put our two moves together. If you can scale vectors and add them, you can build a linear combination: pick some vectors, multiply each by a scalar, and add up the results.

The scalars a and b are the weights. Different weights reach different points. This one little expression is, no exaggeration, the central idea of the whole subject — almost everything ahead is a question about linear combinations in disguise.

Mixing two vectors

Below, \vec{u} and \vec{v} are fixed. Turn the two weight dials and watch a\,\vec{u} + b\,\vec{v} roam the plane. The faint guides show the scaled pieces a\,\vec{u} and b\,\vec{v}; the bold arrow is their sum.

Where can you reach?

Try to land the bold arrow on a few different points. With these two \vec{u} and \vec{v} you can reach every point in the plane — there's exactly one pair of weights for each target. That "reach" has a name, span, and it's the gateway to the next stage. (A warning for later: if \vec{u} and \vec{v} happen to point along the same line, their combinations are stuck on that line and can't reach the whole plane.)