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Version 5 |
| Let $V$ be a vector space over a field $K$. We say that $V$ is |
Let $V$ be a vector space over a field $K$. We say that $V$ is |
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\emph{finite-dimensional} if there exists a finite basis of $V$. Otherwise we
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finite-dimensional if there exists a finite basis of $V$. Otherwise we
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call $V$ \emph{infinite-dimensional}.
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call $V$ infinite-dimensional.
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| It can be shown that every basis of $V$ has the same cardinality. We call this cardinality the dimension of $V$. In particular, if |
It can be shown that every basis of $V$ has the same cardinality. We call this cardinality the 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 $V$. |
$V$ 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 $V$. |
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| Next, let $U\subset V$ a subspace. The dimension of the quotient |
Next, let $U\subset V$ a subspace. The dimension of the quotient |
| vector space $V/U$ is called the codimension of $U$ relative to $V$. |
vector space $V/U$ is called the codimension of $U$ relative to $V$. |
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| Note: in circumstances where the choice of field is ambiguous, the |
Note: in circumstances where the choice of field is ambiguous, the |
| dimension of a vector space depends on the choice of field. For |
dimension of a vector space depends on the choice of field. For |
| example, every complex vector space is also a real vector space, and |
example, every complex vector space is also a real vector space, and |
| therefore has a real dimension, double its complex dimension. |
therefore has a real dimension, double its complex dimension. |
