Difference between revisions of "Linear map"
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==Categories== | ==Categories== | ||
The set of all linear maps from {{M|(U,F)}} to {{M|(V,F)}} is often denoted by <math>\mathcal{L}(U,V)</math> or <math>\text{Hom}(U,V)</math> | The set of all linear maps from {{M|(U,F)}} to {{M|(V,F)}} is often denoted by <math>\mathcal{L}(U,V)</math> or <math>\text{Hom}(U,V)</math> | ||
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| + | ==See also== | ||
| + | [[Example comparing bilinear to linear maps]] | ||
{{Definition|Linear Algebra}} | {{Definition|Linear Algebra}} | ||
Revision as of 15:47, 7 March 2015
Definition
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)
Notations
Some authors use L for a linear map.
Because linear maps can often (always if U and V are finite dimensional) be represented as a matrix sometimes the notation Tv is used instead of T(v)
Homomorphism and isomorphism
A linear map is a vector space homomorphism, if it is a bijection then it is a vector space isomorphism.
Categories
The set of all linear maps from (U,F) to (V,F) is often denoted by L(U,V) or Hom(U,V)
