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Homedimension (vector space)

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# dimension (vector space)

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
call $V$ *infinite-dimensional*.

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 $V$.

Next, let $U\subset V$ a subspace. The dimension of the quotient
vector space $V/U$ is called the *codimension* of $U$ relative to $V$.

Defines:

dimension, codimension, finite-dimensional, infinite-dimensional

Related:

dimension3

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

15A03*no label found*

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