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 (U,F) and (V,F) (it is important that they are over the same field) we say that a map, T:(U,F)→(V,F) or simply T:U→V (because mathematicians are lazy), is a linear map if:
- ∀λ,μ∈F and ∀x,y∈U we have T(λx+μy)=λT(x)+μT(y)
Which is eqivalent to the following:
- T(x+y)=T(x)+T(y)
- T(λx)=λT(x)
Or indeed:
- T(x+λy)=T(x)+λT(y)
