Independence, basis & rank

dimension

The dimension of a space is the number of vectors in any basis for it. Remarkably, no matter which basis you choose, you always need the same count, so this number is a fixed, honest property of the space itself.

Intuitively it is the count of independent directions you have to move in: how many separate knobs you can turn before you can reach everything. A line is 1-dimensional (one knob), a flat plane is 2-dimensional (two knobs), ordinary space is 3-dimensional. The space R^n has dimension n.

Dimension also caps your freedom: in a 2-dimensional space, any 3 vectors must be dependent, because you simply cannot have three genuinely independent directions in a plane.

dim(line)=1, dim(plane)=2, dim(R^3)=3, dim(R^n)=n

Each step up adds one new independent direction to move in.

Dimension counts independent directions; it has nothing to do with how long a list of numbers looks until you check the independence.

Also called

dimensionality维数维度維數維度