2024-09-22

775: Bijective Linear Map Between Vectors Spaces Is 'Vectors Spaces - Linear Morphisms' Isomorphism

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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


  • 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:
F: { the fields }
V1: { the not necessarily finite-dimensional F vectors spaces }
V2: { the not necessarily finite-dimensional F vectors spaces }
f: :V1V2, { the linear maps }{ the bijections }
//

Statements:
f{ the 'vectors spaces - linear morphisms' isomorphisms }
//


2: Natural Language Description


For any field, F, any not necessarily finite-dimensional vectors spaces, V1,V2, and any bijective linear map, f:V1V2, f is a 'vectors spaces - linear morphisms' isomorphism.


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, f1:V2V1; Step 2: see that f1 is linear; Step 3: conclude the proposition.

Step 1:

As f is bijective, the inverse, f1:V2V1, is well-defined.

Step 2:

Let us see that f1 is linear.

Let v1,v2V2 and r1,r2F be any. f1(r1v1+r2v2)=r1f1(v1)+r2f1(v2)? f(r1f1(v1)+r2f1(v2))=r1f(f1(v1))+r2f(f1(v2))=r1v1+r2v2, which means that f1(r1v1+r2v2)=r1f1(v1)+r2f1(v2). So, f1 is linear.

Step 3:

So, f is a 'vectors spaces - linear morphisms' isomorphism.


References


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