linear transformation (original) (raw)

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

• •
  T⁢(v+w)=T⁢(v)+T⁢(w) for all v,w∈V
• •
  T⁢(λ⁢v)=λ⁢T⁢(v) for all v∈V, and λ∈F

The set of all linear maps V→W is denoted by HomF⁡(V,W) or ℒ⁢(V,W).

Properties:

• •
  T⁢(0)=0.
• •
  If S and T are linear transformations from V to W, and k∈F, then so are S+T and k⁢T. As a result, HomF⁡(V,W) is a vector space over F.
• •
  If G:W→U is a linear transformations then G∘T:V→U is also a linear transformation.
• •
  The kernel (http://planetmath.org/KernelOfALinearTransformation)Ker⁡(T)={v∈V∣T⁢(v)=0} is a subspace of V.
• •
  The image (http://planetmath.org/ImageOfALinearTransformation) Im⁡(T)={T⁢(v)∣v∈V} is a subspace of W.
• •
  The inverse image T-1⁢(w) is a subspace if and only if w=0.
• •
  A linear transformation is injective if and only if Ker⁡(T)={0}.
• •
  If v∈V then T-1⁢(T⁢(v))=v+Ker⁡(T).
• •
  If w∈Im⁡(T) then T⁢(T-1⁢(w))={w}.

See also:

• •
  Wikipedia, http://www.wikipedia.org/wiki/Linear\_transformationlinear transformation