definition of general linear group of vectors space
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
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.
Target Context
- The reader will have a definition of general linear group of vectors space.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( V\): \(\in \{\text{ the vectors spaces }\}\)
\(*GL (V)\): \(= \{f: V \to V \vert f \in \{\text{ the linear maps }\} \cap \{\text{ the bijections }\}\}\)
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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: Natural Language Description
For any vectors space, \(V\), the group with the set of all the linear bijections from \(V\) onto \(V\) and the group operation: \((f_1, f_2) \mapsto f_1 \circ f_2\), denoted as \(GL (V)\)
3: 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 \(f_1^{-1} \bullet f_1 = f_1 \bullet f_1^{-1} = id\).
