For every vector space there exists a basis,^{} and all bases of a vector space have equal cardinality;^{} as a result, the dimension of a vector space is uniquely defined. We say *V* is **finite-dimensional** if the dimension of *V* is finite, and **infinite-dimensional** if its dimension is infinite.

The dimension of the vector space *V* over the field *F* can be written as dim_{F}(*V*) or as , read "dimension of *V* over *F*". When *F* can be inferred from context, dim(*V*) is typically written.

The vector space **R**^{3} has

as a basis, and therefore we have dim_{R}(**R**^{3}) = 3. More generally, dim_{R}(**R**^{n}) = *n*, and even more generally, dim_{F}(*F*^{n}) = *n* for any field *F*.

The complex numbers **C** are both a real and complex vector space; we have dim_{R}(**C**) = 2 and dim_{C}(**C**) = 1. So the dimension depends on the base field.

This page was last edited on 4 October 2017, at 21:06.

Reference: https://en.wikipedia.org/wiki/Dimension_of_a_vector_space under CC BY-SA license.

Reference: https://en.wikipedia.org/wiki/Dimension_of_a_vector_space under CC BY-SA license.

- Mathematics
- Vector Space
- Cardinality
- Basis
- Field
- Georg Hamel
- Dimension
- Finite
- Infinite
- Basis
- Field
- Complex Numbers

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