linear transformation
Let V and W be vector spaces over the same field F. A linear transformation is a function T:V→W such that:
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T(v+w)=T(v)+T(w) for all v,w∈V
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T(λv)=λT(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).
Examples:
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Let V=ℝn and W=ℝm and A is any m×n matrix. Then the function LA:V→W defined by LA(v)=Av, the multiplication of matrix A and the vector v (considered as an n×1 matrix), is a linear transformation.
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Let V be the space of all differentiable functions over ℝ and W the space of all continuous functions
over ℝ. Then D:V→W defined by D(f)=f′, the derivative
of f, is a linear transformation.
Properties:
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T(0)=0.
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If S and T are linear transformations from V to W, and k∈F, then so are S+T and kT. As a result, HomF(V,W) is a vector space over F.
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If G:W→U is a linear transformations then G∘T:V→U is also a linear transformation.
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The kernel (http://planetmath.org/KernelOfALinearTransformation) Ker(T)={v∈V∣T(v)=0} is a subspace
of V.
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The image (http://planetmath.org/ImageOfALinearTransformation) Im(T)={T(v)∣v∈V} is a subspace of W.
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The inverse image T-1(w) is a subspace if and only if w=0.
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A linear transformation is injective
if and only if Ker(T)={0}.
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If v∈V then T-1(T(v))=v+Ker(T).
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If w∈Im(T) then T(T-1(w))={w}.
Remark. A linear transformation T:V→W such that W=V is called a linear operator, and a linear functional when W=F.
See also:
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Wikipedia, http://www.wikipedia.org/wiki/Linear_transformationlinear 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 |