Editing Linear map (section)

==Algebraic classifications of linear transformations==

No classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.

Let {{mvar|V}} and {{mvar|W}} denote vector spaces over a field {{mvar|F}} and let {{math|''T'': ''V'' → ''W''}} be a linear map.

===Monomorphism===
{{mvar|T}} is said to be ''[[injective]]'' or a ''[[monomorphism]]'' if any of the following equivalent conditions are true:
# {{mvar|T}} is [[injective|one-to-one]] as a map of [[set (mathematics)|sets]].
# {{math|1=ker ''T'' = {0<sub>''V''</sub>} }}
# {{math|1=dim(ker ''T'') = 0}}
# {{mvar|T}} is [[monic morphism|monic]] or left-cancellable, which is to say, for any vector space {{mvar|U}} and any pair of linear maps {{math|''R'': ''U'' → ''V''}} and {{math|''S'': ''U'' → ''V''}}, the equation {{math|1=''TR'' = ''TS''}} implies {{math|1=''R'' = ''S''}}.
# {{mvar|T}} is [[inverse (ring theory)|left-invertible]], which is to say there exists a linear map {{math|''S'': ''W'' → ''V''}} such that {{math|''ST''}} is the [[Identity function|identity map]] on {{mvar|V}}.

===Epimorphism===
{{mvar|T}} is said to be ''[[surjective]]'' or an ''[[epimorphism]]'' if any of the following equivalent conditions are true:
# {{mvar|T}} is [[surjective|onto]] as a map of sets.
# {{math|1=[[cokernel|coker]] ''T'' = {0<sub>''W''</sub>} }}
# {{mvar|T}} is [[epimorphism|epic]] or right-cancellable, which is to say, for any vector space {{mvar|U}} and any pair of linear maps {{math|''R'': ''W'' → ''U''}} and {{math|''S'': ''W'' → ''U''}}, the equation {{math|1=''RT'' = ''ST''}} implies {{math|1=''R'' = ''S''}}.
# {{mvar|T}} is [[inverse (ring theory)|right-invertible]], which is to say there exists a linear map {{math|''S'': ''W'' → ''V''}} such that {{math|''TS''}} is the [[Identity function|identity map]] on {{mvar|W}}.

===Isomorphism<span class="anchor" id="isomorphism"></span>===
{{mvar|T}} is said to be an ''[[isomorphism]]'' if it is both left- and right-invertible. This is equivalent to {{mvar|T}} being both one-to-one and onto (a [[bijection]] of sets) or also to {{mvar|T}} being both epic and monic, and so being a [[bimorphism]].
{{pb}}
If {{math|''T'': ''V'' → ''V''}} is an endomorphism, then:
* If, for some positive integer {{mvar|n}}, the {{mvar|n}}-th iterate of {{mvar|T}}, {{math|''T''<sup>''n''</sup>}}, is identically zero, then {{mvar|T}} is said to be [[nilpotent]].
* If {{math|1=''T''<sup>2</sup> = ''T''}}, then {{mvar|T}} is said to be [[idempotent]]
* If {{math|1=''T'' = ''kI''}}, where {{mvar|k}} is some scalar, then {{mvar|T}} is said to be a scaling transformation or scalar multiplication map; see [[scalar matrix]].