Difference between revisions of "Linear map"
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==Definition== | ==Definition== | ||
Given two [[Vector space|vector spaces]] {{M|(U,F)}} and {{M|(V,F)}} (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|mathematicians are lazy]]), is a linear map if: | Given two [[Vector space|vector spaces]] {{M|(U,F)}} and {{M|(V,F)}} (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|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> | |
| − | <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> | ||
==Notations== | ==Notations== | ||
Revision as of 17:33, 23 April 2015
Contents
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]
Notations
Some authors use [math]L[/math] for a linear map.
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]
