properties of an affine transformation

In this entry, we prove some of the basic properties of affine transformations. Let α:A1A2 be an affine transformationPlanetmathPlanetmath and [α]:V1V2 its associated linear transformation.

Proposition 1.

α is one-to-one iff [α] is.


Next, suppose α is one-to-one, and T(v)=0 for some vV1. Let P,QA1 with f1(P,Q)=v. Then 0=[α](v)=[α](f1(P,Q))=f1(α(P),α(Q)), which implies that α(P)=α(Q), and therefore P=Q by assumptionPlanetmathPlanetmath. Conversely, suppose [α] is one-to-one, and α(P)=α(Q). Then [α](f1(P,Q))=f2(α(P),α(Q))=0, so that f1(P,Q)=0, and consequently P=Q, showing that α is one-to-one. ∎

Proposition 2.

α is onto iff [α] is.


Suppose α is onto. Let wV2, so there are X,YA2 such that f2(X,Y)=w. Since α is onto, there are P,QA1 with α(P)=X and α(Q)=Y. So w=f2(X,Y)=f2(α(P),α(Q))=[α](f1(P,Q)). Hence [α] is onto. Conversely, assume [α] be onto, and pick YA2. Take an arbitrary point PA1 and set X=α(P). There is vV1 such that [α](v)=f2(X,Y), since [α] is onto. Let QA1 such that f1(P,Q)=v. Then f2(X,α(Q))=f2(α(P),α(Q))=[α](f1(P,Q))=[α](v)=f2(X,Y). But f2(X,-) is a bijectionMathworldPlanetmath, we must have Y=α(Q), showing that α is onto. ∎

Corollary 1.

α is a bijection iff [α] is.

Proposition 3.

A bijectiveMathworldPlanetmath affine transformation α:A1A2 is an affine isomorphism.


Suppose an affine transformation α:A1A2 is a bijection. We want to show that α-1:A2A1 is an affine transformation. Pick any X,YA2, then


By the corollary above, [α] is bijective, and hence a linear isomorphism. So


This shows that α-1 is an affine transformation whose assoicated linear transformation is [α]-1. ∎

Proposition 4.

Two affine spaces associated with the same vector spaceMathworldPlanetmath V are affinely isomorphic.


In fact, all we need to do is to show that (A,f) is isomorphicPlanetmathPlanetmathPlanetmath to (V,g), where g is given by g(v,w)=w-v. Pick any PA, then α:=f(P,-):AV is a bijection. For any vV, there is a unique QA such that v=f(P,Q). Then 1V(f(X,Y))=f(X,Y)=f(P,Y)-f(P,X)=α(Y)-α(X)=g(α(X),α(Y)), showing that 1V is the associated linear transformation of α. ∎

Proposition 5.

Any affine transformation is a linear transformation between the corresponding induced vector spaces. In other words, if α:AB is affine, then α:APBα(P) is linear.


Suppose Q,R,SA are such that Q+R=S, or f1(P,Q)+f1(P,R)=f1(P,S). Then

f2(α(P),α(S)) = [α](f1(P,S))
= [α](f1(P,Q)+f1(P,R))
= [α](f1(P,Q))+[α](f1(P,R))
= f2(α(P),α(Q))+f2(α(P),α(R)),

which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to α(Q)+α(R)=α(S)=α(Q+R).

Next, suppose dQ=R, or df1(P,Q)=f1(P,R), where dD. Then

f2(α(P),α(R)) = [α](f1(P,R))
= [α](df1(P,Q))
= d[α](f1(P,Q))
= df2(α(P),α(Q)),

which is equivalent to dα(Q)=α(R)=α(dQ). ∎

Proposition 6.

If (V,f) is an affine space associated with the vector space V, then the direction f is given by f(v,w)=T(w-v) for some linear isomorphism (invertible linear transformation) T.


By propositionPlanetmathPlanetmath 4, (V,f) is affinely isomorphic to (V,g) with g(v,w)=w-v. Suppose α:(V,f)(V,g) is the affine isomorphism. Then [α](f(v,w))=g(α(v),α(w))=α(w)-α(v). Since [α] is a linear isomorphism, f(v,w)=[α]-1(α(w))-[α]-1(α(v)). By proposition 5, α itself is linear, so f(v,w)=([α]-1α)(w-v). Set T=[α]-1α. Then T is linear and invertiblePlanetmathPlanetmath since α is, our assertion is proved. ∎

Title properties of an affine transformation
Canonical name PropertiesOfAnAffineTransformation
Date of creation 2013-03-22 18:31:53
Last modified on 2013-03-22 18:31:53
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 51A10
Classification msc 15A04
Classification msc 51A15