A description/proof of that for linear map from finite dimensional vectors space, there is domain subspace that is 'vectors Spaces - linear morphisms' isomorphic to image by restriction of map
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of 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 linear map from any finite dimensional vectors space 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 image 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: Description
For any finite dimensional vectors space, \(V_1\), any vectors space, \(V_2\), and any linear map, \(f: V_1 \rightarrow V_2\), there is a domain subspace that is 'vectors spaces - linear morphisms' isomorphic to the map image, \(f (V_1)\), by the restriction of \(f\) on the subspace domain.
2: Proof
By the the proposition that the image of any linear map from any finite dimensional vectors space is a vectors space, \(f (V_1)\) is a vectors space. As is shown in the proof of the proposition, there is a subset of any basis for \(V_1\), \(b_1, b_2, . . ., b_r\), whose space surjectively maps to \(f (V_1)\). As the domain subspace is of the same dimension with \(f (V_1)\), 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, the linear surjection from the domain subspace to \(f (V_1)\) (the restriction of \(f\) to the domain subspace) is a 'vectors spaces - linear morphisms' isomorphism.
