description/proof of that for linear map from finite-dimensional vectors space, there is domain subspace that is 'vectors Spaces - linear morphisms' isomorphic to range by restriction of map
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: Proof
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 %category name% isomorphism.
- The reader admits the proposition that any linear surjection from any finite-dimensional vectors space to any same-dimensional vectors space is a 'vectors spaces - linear morphisms' isomorphism.
- The reader admits the proposition that the image of any finite-dimensional vectors space under any linear map is a vectors space.
Target Context
- The reader will have a description and a proof of the proposition that for any linear map from any finite-dimensional vectors space, there is a domain subspace that is 'vectors spaces - linear morphisms' isomorphic to the map range by the restriction of the map on the subspace domain.
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:
\(F\): \(\in \{\text{ the fields }\}\)
\(V_1\): \(\in \{\text{ the finite-dimensional } F \text{ vectors spaces }\}\)
\(V_2\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\(f\): \(V_1 \to V_2\), \(\in \{\text{ the linear maps }\}\)
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Statements:
\(\exists V \in \{\text{ the subspaces of } V_1\} (f \vert_V: V \to f (V_1) \in \{\text{ the 'vectors spaces - linear morphisms' isomorphisms }\})\)
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2: Natural Language Description
For any field, \(F\), any finite-dimensional \(F\) vectors space, \(V_1\), any \(F\) vectors space, \(V_2\), and any linear map, \(f: V_1 \to V_2\), there is a domain subspace, \(V \subseteq V_1\), that is 'vectors spaces - linear morphisms' isomorphic to the map range, \(f (V_1)\), by the restriction of \(f\) on the \(V\) domain.
3: Proof
Whole Strategy: Step 1: see that \(f (V_1)\) is a vectors space; Step 2: find a subset of any basis for \(V_1\) whose space, \(V\), surjectively maps to \(f (V_1)\); Step 3: see that \(V\) is same-dimensional with \(f (V_1)\); Step 4: conclude the proposition.
Step 1:
By the proposition that the range of any linear map from any finite-dimensional vectors space is a vectors space, \(f (V_1)\) is a vectors space.
Step 2:
In fact, the proof of the proposition that the range of any linear map from any finite-dimensional vectors space is a vectors space has found a subset of any basis for \(V_1\), \(\{b_1, b_2, . . ., b_r\}\), whose space, \(V\), surjectively maps to \(f (V_1)\).
Step 3:
In fact, the proof of the proposition that the range of any linear map from any finite-dimensional vectors space is a vectors space has shown that \(V\) is same-dimensional with \(f (V_1)\).
Step 4:
By the proposition that any linear surjection from any finite-dimensional vectors space to any same-dimensional vectors space is a 'vectors spaces - linear morphisms' isomorphism, \(f \vert_V: V \to f (V_1)\) is a 'vectors spaces - linear morphisms' isomorphism.
