Finite-Dimensional Vector Space¶

Definition¶

A vector space $V$ over a field $\mathbb{F}$ is finite-dimensional if it has a finite spanning list: there are vectors $\boldsymbol{v}_1,\ldots,\boldsymbol{v}_n\in V$ such that

V=\operatorname{span}(\boldsymbol{v}_1,\ldots,\boldsymbol{v}_n).

Equivalently, every $\boldsymbol{v}\in V$ is a linear combination $\boldsymbol{v}=a_1\boldsymbol{v}_1+\cdots+a_n\boldsymbol{v}_n$ with coefficients in $\mathbb{F}$ . The zero vector space is included and has dimension zero. A vector space that is not finite-dimensional is called infinite-dimensional.

Interpretation¶

A vector space is finite-dimensional when finitely many vectors are enough to build every vector in it by linear combinations. This does not mean that the space contains only finitely many vectors: $\mathbb{R}^2$ contains infinitely many vectors, but the two coordinate directions $(1,0)$ and $(0,1)$ span it.

The condition lets the theory work with finite lists. A finite spanning list can be reduced to a basis by removing redundant vectors one at a time, and every basis then has the same finite length, the dimension. So each vector is described by a fixed, finite number of coordinates. The spaces $\mathbb{R}^n$ and $\mathbb{C}^n$ are finite-dimensional, and so is the space of polynomials of degree at most $m$ , which is spanned by $(1,x,\ldots,x^m)$ .

Infinite-dimensional spaces exist as well. All real polynomials, of any degree, form a vector space. The polynomials in a finite list have degree at most some number $d$ , and so has every linear combination of them; the polynomial $x^{d+1}$ is never reached. Hence no finite list spans this space.

Key Results¶

A finite spanning list can be reduced to a basis.
Relative to a basis, every vector has a unique expression as a linear combination of the basis vectors.

Examples¶

The coordinate vectors $(1,0)$ and $(0,1)$ span $\mathbb{R}^2$ .
The space $\mathcal{P}_2(\mathbb{R})$ of polynomials of degree at most two is spanned by $(1,x,x^2)$ .

Prerequisites¶

Vector Space

Sources¶

Aliases: finite dimensional space Symbols: complex numbers, element of, real numbers

Last updated: 20 October 2026 03:02