48: General Linear Group of Vectors Space

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definition of general linear group of vectors space

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of %field name% vectors space.
• The reader knows a definition of group.
• The reader knows a definition of linear map.
• The reader knows a definition of bijection.

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Target Context

• The reader will have a definition of general linear group of vectors space.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$F$: $\in \{\text{ the fields }\}$
$V$: $\in \{\text{ the }F\text{ vectors spaces }\}$
$\ast GL(V)$: $=\{f:V\to V|f\in \{\text{ the linear maps }\}\cap \{\text{ the bijections }\}\}$, $\in \{\text{ the groups }\}$
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Conditions:
$GL(V)$ has the group operation: $:(f_1,f_2)\mapsto f_1\circ f_2$
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2: Note

It is indeed a group, because for any elements, $f_1,f_2,f_3\in GL(V)$, 1) $(f_1\bullet f_2)\bullet f_3=f_1\bullet (f_2\bullet f_3)$; 2) the identity map, $id:V\to V$, is in $GL(V)$ and is the identity element, because $id\bullet f_1=f_1\bullet id=f_1$; 3) the inverse, $f_1^{-1}:V\to V$, of $f_1$ is in $GL(V)$ and is the inverse element of $f_1$, because $f_1^{-1}\bullet f_1=f_1\bullet f_1^{-1}=id$.

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References

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