Difference between revisions of "Linear map"
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| + | Also known as: '''linear transform''' | ||
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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: | ||
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==Notations== | ==Notations== | ||
| − | Some authors use <math>L</math> for a linear map | + | Given a linear map {{M|T}} it can be cumbersome to write {{M|T(v)}} over and over again, so quite often we will write: |
| + | : <math>Tv</math> to mean <math>T(v)</math> | ||
| + | We will fall back to using brackets where needed though, for example: | ||
| + | : <math>T(u+v)</math> being written as <math>Tu+v</math> doesn't work, of course one may write <math>Tu+Tv</math> by the property linear maps are defined to have | ||
| + | ===Common letters used=== | ||
| + | Some authors use <math>L</math> for a linear map, others use {{M|\tau}} | ||
Because linear maps can often (always if {{M|U}} and {{M|V}} are finite dimensional) be represented as a [[Matrix|matrix]] sometimes the notation <math>Tv</math> is used instead of <math>T(v)</math> | Because linear maps can often (always if {{M|U}} and {{M|V}} are finite dimensional) be represented as a [[Matrix|matrix]] sometimes the notation <math>Tv</math> is used instead of <math>T(v)</math> | ||
Revision as of 19:15, 24 April 2015
Also known as: linear transform
Contents
[hide]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)
Which is eqivalent to the following:
- T(x+y)=T(x)+T(y)
- T(λx)=λT(x)
Notations
Given a linear map T it can be cumbersome to write T(v) over and over again, so quite often we will write:
- Tv to mean T(v)
We will fall back to using brackets where needed though, for example:
- T(u+v) being written as Tu+v doesn't work, of course one may write Tu+Tv by the property linear maps are defined to have
Common letters used
Some authors use L for a linear map, others use τ
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)
Between a basis
The Change of basis matrix ought to be denoted [Id]BA where A is the source basis and B is the target, see this page for a tour of notation and the use of [⋅]BA
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 (U,F) to (V,F) is often denoted by L(U,V) or Hom(U,V)
