Dimension¶

Definition¶

The dimension of a finite-dimensional vector space is the length of any basis of that vector space. It is denoted by $\dim(V)$ .

Interpretation¶

Dimension counts how many independent directions are needed to describe every vector in the space: it is the number of vectors in a basis, and so the number of coordinates each vector gets. The definition makes sense because in a finite-dimensional vector space all bases have the same length.

Three coordinate axes with a shaded plane through the origin of dimension 2, spanned by two arrows, and a line through the origin of dimension 1, spanned by one arrow.

The figure shows the three cases inside $\mathbb{R}^3$ . One nonzero arrow spans a line through the origin, of dimension $1$ . Two arrows that do not lie on one line span a plane through the origin, of dimension $2$ . Three independent arrows are needed for all of $\mathbb{R}^3$ , of dimension $3$ . The line and the plane are subspaces of $\mathbb{R}^3$ with their own dimension, not the dimension $3$ of the surrounding space.

Dimension can be read in two equivalent ways: it is the largest number of linearly independent vectors in $V$ , and the smallest number of vectors in a spanning list of $V$ . So in $\mathbb{R}^2$ any three vectors are linearly dependent, and no single vector spans $\mathbb{R}^2$ . In the same way, a subspace never has a larger dimension than the space that contains it.

Dimension is not the number of vectors in the space: the line in the figure contains infinitely many vectors but has dimension $1$ . The zero space $\{\boldsymbol{0}\}$ is the only space of dimension $0$ .

Key Results¶

Any two bases of a finite-dimensional vector space have the same length.
Every basis of $\mathbb{R}^n$ contains exactly $n$ vectors.

Examples¶

$\mathbb{R}^2$ has dimension $2$ .
The vector space of polynomials of degree at most $2$ has dimension $3$ , with basis $(1,x,x^2)$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 2
Math V: Advanced Linear Algebra · Week 5 (revisited)

Aliases: Hamel dimension, vector space dimension Keywords: dimension of a vector space Symbols: real numbers

Last updated: 20 October 2026 03:02