definition of rank of linear map between vectors spaces
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of linear map.
- The reader knows a definition of dimension of vectors space.
Target Context
- The reader will have a definition of rank of linear map between vectors spaces.
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:
\( F\): \(\in \{\text{ the fields }\}\)
\( V_1\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\( V_2\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\( f\): \(: V_1 \to V_2\), \(\in \{\text{ the linear maps }\}\)
\(*Rank (f)\): \(= Dim (f (V_1))\), the dimension of \(f (V_1)\)
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Conditions:
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2: Note
It is well-defined, because \(f (V_1)\) is a vectors space, by the proposition that the range of any linear map between any vectors spaces is a vectors subspace of the codomain.
When \(f (V_1)\) is infinite-dimensional, \(Rank (f)\) may not be any natural number but a cardinal number.
