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'dimension (vector space)'
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| Title of object: |
dimension (vector space) |
| Canonical Name: |
Dimension2 |
| Type: |
Definition |
| Created on: |
2002-06-01 23:32:50 |
| Modified on: |
2006-09-06 17:39:03 |
| Classification: |
msc:15A03 |
| Defines: |
dimension, codimension, finite-dimensional, infinite-dimensional |
Revision comment (for changes between this and next version):
| Changes for correction #9610 ('emphasis on defined terms'). |
Preamble:
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Content:
Let $V$ be a vector space over a field $K$. We say that $V$ is
\emph{finite-dimensional} if there exists a finite basis of $V$. Otherwise we
call $V$ \emph{infinite-dimensional}.
It can be shown that every basis of $V$ has the same cardinality. We call this cardinality the \emph{dimension} of $V$. In particular, if
$V$ is finite-dimensional, then every basis of $V$ will consist of a finite set $v_1,\ldots, v_n$. We then call the natural number $n$ the dimension of $V$.
Next, let $U\subset V$ a subspace. The dimension of the quotient
vector space $V/U$ is called the codimension of $U$ relative to $V$.
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. |
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