linear transformation

Let V and W be vector spacesMathworldPlanetmath over the same field F. A linear transformation is a function T:VW such that:

  • T(v+w)=T(v)+T(w) for all v,wV

  • T(λv)=λT(v) for all vV, and λF

The set of all linear maps VW is denoted by HomF(V,W) or (V,W).


  • Let V=n and W=m and A is any m×n matrix. Then the function LA:VW defined by LA(v)=Av, the multiplication of matrix A and the vector v (considered as an n×1 matrix), is a linear transformation.

  • Let V be the space of all differentiable functions over and W the space of all continuous functionsMathworldPlanetmathPlanetmath over . Then D:VW defined by D(f)=f, the derivativeMathworldPlanetmathPlanetmath of f, is a linear transformation.


  • T(0)=0.

  • If S and T are linear transformations from V to W, and kF, then so are S+T and kT. As a result, HomF(V,W) is a vector space over F.

  • If G:WU is a linear transformations then GT:VU is also a linear transformation.

  • The kernel ( Ker(T)={vVT(v)=0} is a subspacePlanetmathPlanetmath of V.

  • The image ( Im(T)={T(v)vV} is a subspace of W.

  • The inverse image T-1(w) is a subspace if and only if w=0.

  • A linear transformation is injectivePlanetmathPlanetmath if and only if Ker(T)={0}.

  • If vV then T-1(T(v))=v+Ker(T).

  • If wIm(T) then T(T-1(w))={w}.

Remark. A linear transformation T:VW such that W=V is called a linear operator, and a linear functionalMathworldPlanetmathPlanetmath when W=F.

See also:

  • Wikipedia, transformation

Title linear transformation
Canonical name LinearTransformation
Date of creation 2013-03-22 11:56:41
Last modified on 2013-03-22 11:56:41
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 24
Author CWoo (3771)
Entry type Definition
Classification msc 15A04
Synonym linear map
Synonym vector space homomorphism
Synonym linear mapping
Related topic Matrix
Related topic InvariantSubspace
Related topic DualHomomorphism
Related topic KernelOfALinearTransformation
Related topic EigenvalueOfALinearOperator
Related topic NilpotentTransformation
Related topic AffineTransformation
Related topic SubLinear
Related topic MatrixRepresentationOfALinearTransformation
Defines linear operator