Difference between revisions of "Linear map/Definition"
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(Created page with "Given two 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...") |
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* <math>T(x+y)=T(x)+T(y)</math> | * <math>T(x+y)=T(x)+T(y)</math> | ||
* <math>T(\lambda x)=\lambda T(x)</math> | * <math>T(\lambda x)=\lambda T(x)</math> | ||
| + | Or indeed: | ||
| + | * <math>T(x+\lambda y)=T(x)+\lambda T(y)</math> | ||
Revision as of 10:33, 12 June 2015
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]
