| Version 3 |
Version 2 |
| Let $V$ be a vector space over a field $K$. We say that $V$ is |
Let $V$ be a vector space. We say that $V$ is finite-dimensional if |
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there exists a finite basis of $V$. Otherwise we call $V$ |
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infinite-dimensional. |
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It can be shown that every basis of $V$ has the same cardinality. If |
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$V$ is finite-dimensional, we call this cardinality the dimension of |
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$V$. |
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Next, let $U\subset V$ a subspace. The dimension of the quotient |
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vector space $V/U$ is called the codimension of $U$ relative to $V$.Let $V$ be a vector space over a field $K$. We say that $V$ is |
| finite-dimensional if there exists a finite basis of $V$. Otherwise we |
finite-dimensional if there exists a finite basis of $V$. Otherwise we |
| call $V$ infinite-dimensional. |
call $V$ infinite-dimensional. |
| It can be shown that every basis of $V$ has the same cardinality. If |
It can be shown that every basis of $V$ has the same cardinality. If |
| $V$ is finite-dimensional, we call this cardinality the dimension of |
$V$ is finite-dimensional, we call this cardinality the dimension of |
| $V$. |
$V$. |
| 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 $V$. |
| 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. |
