MATH2701

Transformation on R²

bijection from R² to itself

Transformation T(A,b)

x->Ax+b provided det(A)≠0

Transformations t,t',t'' must adhere to

1. closure - t°t' is a transformation
2. associativity - t°(t'°t'')=(t°t')°t''
3. identity - t°1=t=1°t
4. inverses - t-¹ is a transformation

Group

A set G equipped with a map *:G×G→G, (g,g')→g*g' w/:
1. (g*g')*g''=g*(g'*g'') for all g,g',g''∈G
2. There exists e∈G such that g*e=g=e*g for all g∈G
3. Every g∈G there exists h∈G such that h*g=e=g*h

The elements e and h are...

unique

The set of all transformations

forms a group

The set of all translations

forms an abelian group

The set of all collineations

forms a group

Abelian

if g*g'=g'*g for all g,g'∈G

Order

a positive integer n such that g^i≠e for i=1,...,n-1 and g^n=e then g has order n. If no such n exists, g has infinite order.

Subgroup

A nonempty subset H of a group G is a subgroup if it forms a group. H≤G

Subgroup Theorem

A nonempty subset H of G is a subgroup iff:
1. for all h,h'∈H we have h*h'∈H
2. for every h∈H we have h^-1∈H

The general linear group GL_n(R)

Matrices A in M_n(R) such that det(A)≠0.

The subgroup generated by the subgroup S

set of all products of elements of S and the inverses of the elements of S

The orthogonal group O_n(R)

A in GL_n(R) such that A^T*A=I

|a|

=sqrt(a.a)

Orthogonal

a.b=0

Angle between a and b

cos(theta)=(a.b)/|a||b|

The projection of a onto b

(a.b)/(b.b)*b

Isometry

a transformation that preserves distances:
d(T(P),T(Q))=d(P,Q) or |T(x)-T(y)|=x-y for all P,Q,x,y

Examples of isometries

Translations, reflections and rotations

The set of all isometries

forms a group

Let Q be orthogonal matrix and b an integer. Then T(Q,b)

is an isometry

Any isometry can be written in the form...

T(Q,b) where Q is an orthogonal matrix and b some vector

Every isometry is a...

collineation

If an isometry fixes the zero vector...

1. then it preserves dot products: T(x).T(y)=x.y
2. T=T(Q,0) where Q is an orthogonal matrix such that each column is given by T(e_1),...

All isometries are...

invertible

Reflection in l where l is a line in R^2

If P∈l then T(P)=P
If P∉l then T(P)=Q where Q is the unique point such that l is the perpendicular bisector of the line segment PQ

For any reflection in l...

T^2=1

Every reflection is...

an isometry

Involution

any map such that T^2=1, it must be a transformation

If an isometry fixes three non-collinear points...

then it is the identity

If an isometry fixes two points...

then either it is the identity or is the reflection in the line through those points

If an isometry fixes one point...

then it is a product of two reflections

Any isometry is...

a product of at most three reflections

An isometry is a translation iff

it is a product of two reflections in parallel lines

A geometrical figure is congruent to another geometrical figure if...

there is an isometry that maps between them such that T(A)=B. A≅B.

Congruence is...

an equivalence relation

A symmetry of a geometrical figure is...

an isometry such that T(A)=A.

The symmetries of a geometrical figure...

forms a subgroup of the isometries.

A rotation about a point C through θ

is the transformation p_(C,θ) that fixes C and sends P to P' where d(C,P)=d(C,P') and CP to CP' has an angle θ. Anticlockwise if θ>0

The rotation p_(0,θ)=Qx where...

Q=(cosθ -sinθ)
(sinθ cosθ)
Q is orthogonal

p(C,θ)=

T_c°p_(0,θ)°T_-c

The set of all rotations about a fixed point C

is an abelian subgroup of the isometries

An isometry is a rotation iff

it is a product of two reflections in intersecting lines.

A halfturn is...

a rotation of the form p_(C,

The point reflection around a point p sends a point x to the point...

2p-x

Even isometry

a product of even number of reflections

No isometry can be...

both even and odd

Let C be a point and let p,q,r be lines through C. Then there exists a line l through C such that...

σ_r σ_p σ_q =σ_l

The set of even isometries...

forms a subgroup of isometries

An isometry is a glide reflection with axis c if...

there exist distinct lines a and b both perpendicular to c such that T=σ_c σ_b σ_a

Let p,q,r be lines. Then the commutation of each of these reflections is a glide reflection iff...

p,q,r are neither concurrent nor parallel

A glide reflection has the following 5 properties:

1) it is a non-identity translation followed by a reflection.
2) fixes no points
3) fixes exactly one line, c
4) is a composite of a reflection in a line a and a halfturn about some point which does not line on a.
5) the midpoint of any point and its image under a glide reflection lies on the axis of the glide reflection

Every plane isometry is exactly one of:

1) the identity
2) a non-identity translation
3) a non-identity rotation
4) a reflection
5) a glide reflection

A transformation is a similarity if...

there exists r>0 such that d(T(P),T(Q))=r.d(P,Q) for all P,Q. We say that it has ratio r.

Every isometry is a...

similarity

Similarities preserve...

angles

Stretch

A stretch of ratio r>0 around a point C is the transformation δ_(C,r) that fixes C and otherwise sends a point P to the point P' where P' is the unique point on the ray through P such that d(C,P')=r.d(C,P).
That is,
c+r(x-c)

The set of all similarities...

forms a subgroup of the transformations.

A stretch is a...

similarity

Let T be a similarity of ratio r>0, and let C be any point. Then there are...

unique isometries σ and σ' such that T can be written as T=σδ_(C,r) and as T=δ_(C,r)σ

Every similarity is...

a collineation

A collineation is a dilatation...

if l || T(l) for every line l.

A dilation is...

either a stretch of ratio r about a point C (δ_(C,r)) or a stretch of ratio r about a point C followed by a point reflection about C (δ_(C,-r).

If T is a translation or a dilation...

then T is a dilatation

Every dilatation is...

either a translation or a dilation

The dilatations...

form a subgroup of the collineations.

Let A,B,A',B' be points w/ A≠B and A'≠B'. Suppose that AB || A'B'. Then there exists...

a unique dilatation δ such that δ(A)=A' and δ(B)=B'

A stretch reflection is...

a non-identity stretch about some point C followed by a reflection in some line through C

Every plane similarity is exactly one of...

1) An isometry
2) A non-identity stretch
3) A stretch reflection
4) A stretch rotation, where the rotation is not the identity

A stretch rotation...

is a non-identity stretch about some point C, followed by a non-identity rotation about C.

Let T be a plane similarity that is not an isometry. Then...

has a fixed point.

Italics I

group of isometries of R^2

italics T

Group of translations of R^2

Italics S

Group of similarities of R^2

Italics D

Group of dilatations of R^2

The group of halfturns is also the intersection of I and D.

Italics H

Group of halfturns union with italics T

Let s be a similarity and t an isometry. Then sts^-1 is...

an isometry

Let s be a similarity and t a translation. Then sts^-1 is...

a translation

Let s be a similarity and t a halfturn (or translation). Then sts^-1 is...

a halfturn (or translation)

Let s be a similarity and t a dilatation. Then sts^-1 is...

a dilatation

Let G be a group. A subgroup K≤G is a normal subgroup if...

gkg^-1 is in K for all g in G and k in K. K≤G (but triangle)

The groups I, T, D and H are...

all normal subgroups of similarities.

Let G and H be groups. A function f:G→H is a group homomorphism if...

f(g_1 g_2)=f(g_1)f(g_2) for all g_1,g_2 in G.

Suppose that K is a normal subgroup of G and g is in G. Then the left coset is...

gK={gk|k∈K}

A group homomorphism is a group isomorphism if...

the function f is a bijection.

Suppose that K is a normal subgroup of G and g is in G. Then the right coset is...

Kg={kg|k∈K}

Both the left and right cosets are...

subsets of G.

G/K is...

the quotient group of G by K which is {gK|g∈G}. That is, the set of all the left cosets with the multiplication (gK)(g'K)=(gg'K)

If K is a normal subgroup of G, the left and right cosets are...

the same.

The image of a function f:G→H is...

{h∈H|f(g)=h for some g∈G}

The kernel of f is...

{g∈G|f(g)=1}

The kernel is...

a normal subgroup

The first isomorphism theorem

Let f:G→H be a group homomorphism with K=ker(f). Then G/K≅im(f).

Let α be a collineation on R^2 and let l and m be parallel lines. Then...

α(l) and α(m) are parallel

Let α be a transformation on R^2. Then α is a collineation iff

for any 3 collinear points A,B,C, the images of α(A),α(B),α(C) are collinear

Darboux's Theorem

Let alpha be a collineation, and let C be a point between points A and B. Let A', B', C' be the images under alpha of A,B,C. Then C' lies between A' and B'.

Let alpha be a collineation that fixes three non-collinear points A,B,C. Then...

alpha is the identity