Linear map
Also known as: linear transform
Contents
[<hidetoc>]Definition
Given two vector spaces (U,F) and (V,F) (it is important that they are over the same field) we say that a map, T:(U,F)→(V,F)
- ∀λ,μ∈Fand ∀x,y∈Uwe have T(λx+μy)=λT(x)+μT(y)
Which is eqivalent to the following:
- T(x+y)=T(x)+T(y)
- T(λx)=λT(x)
Or indeed:
- T(x+λy)=T(x)+λT(y)[1]
Terminology
Map types:
Type | Description |
---|---|
Linear map | Synonym for linear transform |
Linear transform | What we'd call linear map, it's just a map T:(V,F)→(W,F) where T(αu+βv)=αT(u)+βT(v) ∀u,v∈V ∀α,β∈F |
Linear operator | A linear transform into the same space as the domain, that is T:(V,F)→(V,F) |
Map terms[2]:
Term | Meaning | Example (T is linear map) |
---|---|---|
Homomorphism | Any linear transform | T:(U,F)→(V,F) |
Endomorphism | Any linear operator | T:(W,F)→(W,F) |
Monomorphism (Embedding) | Any injective linear transform | T:(U,F)→(V,F) where T is injective |
Epimorphism | Any surjective linear transform | T:(U,F)→(V,F) where T is surjective |
Isomorphism | Any bijective linear transform | T:(U,F)→(V,F) where T is a bijection |
Automorphism | Any bijective linear operator | T:(W,F)→(W,F) where T is a bijection |
Notations
Given a linear map T it can be cumbersome to write T(v) over and over again, so quite often we will write:
- Tvto mean T(v)
We will fall back to using brackets where needed though, for example:
- T(u+v)being written as Tu+vdoesn't work, of course one may write Tu+Tvby the property linear maps are defined to have
Common letters used
Some authors use L
Because linear maps can often (always if U and V are finite dimensional) be represented as a matrix sometimes the notation Tv
Between a basis
The Change of basis matrix ought to be denoted [Id]BA
Homomorphism, isomorphism and isometry
A linear map is a vector space homomorphism, if it is a bijection then it is invertible, but the word isomorphism should be used sparingly, to avoid confusion with linear isometries which ought to be called "isometries"
Using the prefix "linear" avoids this, eg:
- Linear homomorphism
- Linear isomorphism
- Linear isometry
Categories
The set of all linear maps from (U,F) to (V,F) is often denoted by L(U,V)
See also
References
- <cite_references_link_accessibility_label> ↑ Linear Algebra via Exterior Products - Sergei Winitzki
- <cite_references_link_accessibility_label> ↑ Advanced Linear Algebra - Third Edition - Steven Roman - Graduate Texts in Mathematics