Dimension

Precise definition

The dimension of a finite-dimensional vector space $V$, written $\dim V$, is the number of vectors in any basis of $V$. The basis theorem guarantees this number is well-defined. The zero space has dimension 0 because its basis is the empty list.

Notation and mathematical language

Examples: $\dim\mathbb R^n=n$; polynomials of degree at most $m$ have basis $1,x,\ldots,x^m$ and dimension $m+1$; $m\times n$ matrices have dimension $mn$. For $T:V\to W$, rank–nullity says $\dim V=\dim\ker T+\dim\operatorname{range}T$ when $V$ is finite-dimensional.

Conceptual picture

Dimension counts independent directions, not the number of vectors in the space. A plane through the origin contains infinitely many vectors but needs two basis directions. Constraints reduce dimension when they are independent.

Fully worked example

Interpretation and application

Dimension predicts degrees of freedom in models, data subspaces, polynomial approximation, and solution sets. It is invariant under a change of basis, so different coordinate descriptions do not change the underlying count.

Common mistakes