Put the two best ideas together. A basis for the plane is a set of vectors that is both

• linearly independent — no waste, every vector pulls a new direction, and
• spanning — between them they reach every point.

A basis is a perfect set of building blocks: just enough vectors to describe everything, with nothing redundant. The standard basis for 2D is \{\hat{\imath}, \hat{\jmath}\}, but it is far from the only one — any two independent vectors will do.

A basis builds a grid

Each basis lays its own coordinate grid over the plane. Tilt the second basis vector below and watch the grid skew with it — yet as long as the two vectors stay independent, the grid still tiles the whole plane, so every point still has coordinates. Make them collinear and the grid collapses: no longer a basis.

Dimension: the magic number

Here is the remarkable fact: every basis for a given space has the same number of vectors. That count is the dimension of the space — 2 for the plane, 3 for the space we live in, and n for the lists of n numbers that machine learning runs on. Dimension is the honest measure of "how many independent directions are really here" — and it never depends on which basis you chose.