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This article has 10 links. View as Cloud or List.
| 1 | Vector Subspace |
| 1 | Vector Space |
| 1 | Complex |
| 1 | Field |
| 1 | Real Number |
| 1 | Finite |
| 1 | Natural Number |
| 1 | Cardinality |
| 2 | Basis |
| 4 | Quotient Module |
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Dimension
Let 
be a vector space
over a field
. We say that 
is
finite-dimensional if there exists a finite
basis
of 
. Otherwise we
call 
infinite-dimensional.
It can be shown that every basis of 
has the same cardinality. We call this cardinality the dimension of 
. In particular, if
is finite-dimensional, then every basis of 
will consist of a finite set
. We then call the natural number
the dimension of 
.
Next, let
a subspace. The dimension of the quotient
vector space
is called the codimension of 
relative to 
.
In circumstances where the choice of field is ambiguous, the dimension of a vector space depends on the choice of field. For example, every complex vector space is also a real vector space, and therefore has a real dimension, double its complex dimension.