Dimension (of a vector space)

Definition · applies to vector spaces

Elementary

The dimension of an object is the number of perpendicular directions you can move within that space. Think of the directions you can move in a space: along a line, you can only go back and forth but not side to side or up and down, so a line has one dimension.

High school

Consider a space, such as a line, a plane, or the space around us, and coordinates that describe each point in it. The dimension of the space is the number of coordinates needed to describe a point.

Undergraduate

The dimension of the vector space $V$ is the minimum number of vectors needed to span $V$, which is also the number of vectors in any basis of $V$.

Examples

Worked examples

• A line : $\dim(\text{line}) = 1$ Elementary
• A flat surface (a plane) : $\dim(\text{plane}) = 2$ Elementary
• $\reals^3$ : $\dim \reals^3 = 3$ High school
  A basis of $\reals^3$ has $3$ vectors.

Notes

Other notions of dimension include the dimension of a manifold, the Hausdorff dimension of a fractal, and the dimension of a physical quantity.

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