- Summary
- Text implies that every one to one linear transformation of dimension n on field F is onto

Schaum's Outline of Group Theory, Section 3.6e defines [itex]{{\rm{L}}_n}\left( {V,F} \right)[/itex] as the set of all one to one linear transformations of V,

the vector space of dimension n over field F.

It then says "[itex]{{\rm{L}}_n}\left( {V,F} \right) \subseteq {S_V}[/itex], clearly".

([itex]{S_V}[/itex] here means the set of all one to one mappings of V onto V).

This isn't clear to me at all.

By the definition given, an element of [itex]{{\rm{L}}_n}\left( {V,F} \right)[/itex] could potentially not be onto V.

Then it wouldn't be an element of [itex]{S_V}[/itex].

Either all such one to one linear transformations have to be onto, or the author should have defined [itex]{{\rm{L}}_n}\left( {V,F} \right)[/itex]

as the set of all one to one linear transformations of V onto V, the vector space of dimension n over F.

I haven't had much luck trying to prove that all such one to one transformations have to be onto, so I am guessing the author made a mistake.

On the next page after this definition, the author calls [itex]{{\rm{L}}_n}\left( {V,F} \right)[/itex], with composition of mappings as the operation,

the full linear group of dimension n. This doesn't seem to be standard terminology so its hard to find anything online to verify my suspicion.

Can anyone verify that the author made a mistake, or show me how to prove that all such one to one transformations have to be onto?

Thanks.

the vector space of dimension n over field F.

It then says "[itex]{{\rm{L}}_n}\left( {V,F} \right) \subseteq {S_V}[/itex], clearly".

([itex]{S_V}[/itex] here means the set of all one to one mappings of V onto V).

This isn't clear to me at all.

By the definition given, an element of [itex]{{\rm{L}}_n}\left( {V,F} \right)[/itex] could potentially not be onto V.

Then it wouldn't be an element of [itex]{S_V}[/itex].

Either all such one to one linear transformations have to be onto, or the author should have defined [itex]{{\rm{L}}_n}\left( {V,F} \right)[/itex]

as the set of all one to one linear transformations of V onto V, the vector space of dimension n over F.

I haven't had much luck trying to prove that all such one to one transformations have to be onto, so I am guessing the author made a mistake.

On the next page after this definition, the author calls [itex]{{\rm{L}}_n}\left( {V,F} \right)[/itex], with composition of mappings as the operation,

the full linear group of dimension n. This doesn't seem to be standard terminology so its hard to find anything online to verify my suspicion.

Can anyone verify that the author made a mistake, or show me how to prove that all such one to one transformations have to be onto?

Thanks.