description/proof of that bijective linear map between vectors spaces is 'vectors spaces - linear morphisms' isomorphism
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
- 4: Proof
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of bijection.
- The reader knows a definition of linear map. The reader knows a definition of %category name% isomorphism.
Target Context
- The reader will have a description and a proof of the proposition that any bijective linear map between any vectors spaces is a 'vectors spaces - linear morphisms' isomorphism.
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:
//
Statements:
//
2: Natural Language Description
For any field,
3: Note
In general, a bijective morphism is not necessarily an isomorphism: for example, a bijective continuous map is not necessarily a 'topological spaces - continuous maps' isomorphism, because the inverse is not necessarily continuous, which is the reason why we need to specifically prove this proposition.
4: Proof
Whole Strategy: Step 1: define the inverse,
Step 1:
As
Step 2:
Let us see that
Let
Step 3:
So,