basis

A basis is a Goldilocks set of vectors for a space: not too few, not too many, just right. It is independent (no vector is wasted) AND it spans the whole space (the vectors together can reach everything). It is the minimal complete kit of building blocks.

The magic payoff is uniqueness. Once you fix a basis, every vector in the space has exactly ONE recipe in terms of those building blocks. Those weights are the vector's coordinates. This is why a basis turns a fuzzy geometric space into clean lists of numbers you can compute with.

Example: in the plane, (1,0) and (0,1) are the familiar basis; the point (4,3) has the unique recipe 4*(1,0) + 3*(0,1). But (1,1) and (1,-1) are also a perfectly good basis for the same plane, just with different coordinates.