Linear map

Also known as: linear transform

Definition

Given two vector spaces [ilmath](U,F)[/ilmath] and [ilmath](V,F)[/ilmath] (it is important that they are over the same field) we say that a map, [math]T:(U,F)\rightarrow(V,F)[/math] or simply [math]T:U\rightarrow V[/math] (because mathematicians are lazy), is a linear map if:

• [math]\forall \lambda,\mu\in F[/math] and [math]\forall x,y\in U[/math] we have [math]T(\lambda x+\mu y) = \lambda T(x) + \mu T(y)[/math]

Which is eqivalent to the following:

• [math]T(x+y)=T(x)+T(y)[/math]
• [math]T(\lambda x)=\lambda T(x)[/math]

Or indeed:

• [math]T(x+\lambda y)=T(x)+\lambda T(y)[/math]

Terminology

Map types:

Type	Description
Linear map	Synonym for linear transform
Linear transform	What we'd call linear map, it's just a map [ilmath]T:(V,F)\rightarrow(W,F)[/ilmath] where [ilmath]T(\alpha u+\beta v)=\alpha T(u)+\beta T(v)\ \forall u,v\in V\ \forall\alpha,\beta\in F[/ilmath]
Linear operator	A linear transform into the same space as the domain, that is [ilmath]T:(V,F)\rightarrow(V,F)[/ilmath]

Map terms:

Term	Meaning	Example ([ilmath]T[/ilmath] is linear map)
Homomorphism	Any linear transform	[ilmath]T:(U,F)\rightarrow(V,F)[/ilmath]
Endomorphism	Any linear operator	[ilmath]T:(W,F)\rightarrow(W,F)[/ilmath]
Monomorphism (Embedding)	Any injective linear transform	[ilmath]T:(U,F)\rightarrow(V,F)[/ilmath] where [ilmath]T[/ilmath] is injective
Epimorphism	Any surjective linear transform	[ilmath]T:(U,F)\rightarrow(V,F)[/ilmath] where [ilmath]T[/ilmath] is surjective
Isomorphism	Any bijective linear transform	[ilmath]T:(U,F)\rightarrow(V,F)[/ilmath] where [ilmath]T[/ilmath] is a bijection
Automorphism	Any bijective linear operator	[ilmath]T:(W,F)\rightarrow(W,F)[/ilmath] where [ilmath]T[/ilmath] is a bijection

Notations

Given a linear map [ilmath]T[/ilmath] it can be cumbersome to write [ilmath]T(v)[/ilmath] over and over again, so quite often we will write:

[math]Tv[/math] to mean [math]T(v)[/math]

We will fall back to using brackets where needed though, for example:

[math]T(u+v)[/math] being written as [math]Tu+v[/math] doesn't work, of course one may write [math]Tu+Tv[/math] by the property linear maps are defined to have

Common letters used

Some authors use [math]L[/math] for a linear map, others use [ilmath]\tau[/ilmath]

Because linear maps can often (always if [ilmath]U[/ilmath] and [ilmath]V[/ilmath] are finite dimensional) be represented as a matrix sometimes the notation [math]Tv[/math] is used instead of [math]T(v)[/math]

Between a basis

The Change of basis matrix ought to be denoted [math][Id]_A^B[/math] where [ilmath]A[/ilmath] is the source basis and [ilmath]B[/ilmath] is the target, see this page for a tour of notation and the use of [math][\cdot]_A^B[/math]

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 [ilmath](U,F)[/ilmath] to [ilmath](V,F)[/ilmath] is often denoted by [math]\mathcal{L}(U,V)[/math] or [math]\text{Hom}(U,V)[/math]

See also

• Multilinear map
• Tensor product
• Example comparing bilinear to linear maps
• Kernel