# Linear map

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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, $T:(U,F)\rightarrow(V,F)$ or simply $T:U\rightarrow V$ (because mathematicians are lazy), is a linear map if:

• $\forall \lambda,\mu\in F$ and $\forall x,y\in U$ we have $T(\lambda x+\mu y) = \lambda T(x) + \mu T(y)$

Which is eqivalent to the following:

• $T(x+y)=T(x)+T(y)$
• $T(\lambda x)=\lambda T(x)$

Or indeed:

• $T(x+\lambda y)=T(x)+\lambda T(y)$

## 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:

$Tv$ to mean $T(v)$

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

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

### Common letters used

Some authors use $L$ 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 $Tv$ is used instead of $T(v)$

### Between a basis

The Change of basis matrix ought to be denoted $[Id]_A^B$ 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 $[\cdot]_A^B$

## 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 $\mathcal{L}(U,V)$ or $\text{Hom}(U,V)$