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4.2 Transformation Geometry

We will denote the Euclidean Plane by E.

Let f be a correspondence that assigns points E to points in E. (so E is both
‘source’ and ‘target’ for f.

Example 1: f ( x, y )  ( x,1/ y )

The set of allowable inputs of the correspondence is called the domain of the
correspondence. In Example 1, the domain of f, D( f )  {( x, y ) | y  0}  E .

If to each point in the domain, there corresponds a unique output, we call the
correspondence a function. f in Example 1 is a function.

The set of outputs of a function is called the range of the function. Observe that
the range of f in Example 1 is R( f )  {( x, y ) | y  0}  E .

A function is said to be one-to-one if different inputs give different outputs, i.e.
(a, b)  (c, d )  f (a, b)  f (c, d )
or contrapositively
f ( a , b )  f ( c , d )  ( a , b )  ( c, d ) .

The function f : X  Y is said to be onto, if for every y  Y , ,  x  X so that
f ( x)  y.

So, the function f : E  E is onto if R( f )  E ( target) .

The function f : E  E which is both one-to-one and onto is called a bijection.

A linear transformation (or operator) T on a vector space V is a correspondence
that assigns to every vector x in V a vector T(x) in V , in such a way that
T (x  y )  T (x)   (y )
for x, y  V &  ,   the underlying scalar field of the vector space.
A new model for the Euclidean Plane ( E ), z  1
Definition: A point x in the Euclidean plane is any ordered-triple of the form ( x, y,1)
where x & y are real numbers.
Definition of Addition in E. For ( x, y,1) and (u , v,1) in E, we define
( x, y,1) + (u , v,1) = ( x  u , y  v,1) .
Definition of Scalar multiplication in E. For ( x, y,1) in E, we define
  ( x, y,1) = (x, y,1) .
Claim: ( E ,,) is a vector space.
A point x  ( x, y,1) in the plane is identified with the 3 1 matrix x
y 1]
T
Consider the map, T : E  E defined by T (x)  Ax
where
a b c 
A  e f g  with A  0
0 0 1
Observe:
a b c   x  ax  by  c 
1.
Ax  e f g   y   ex  fy  g   E
0 0 1 1  

1
2.
3.
4.
5.
T is one – to – one (prove)
T is onto (prove)
T is has a unique inverse (prove)
T is a linear transformation (prove)
Theorem:
The image of a line L  u
v
a
w under a linear transformation A  e
0
b
f
0
c
g 
1
is given by LA 1  kL'
Exercises
1.
Given a linear transformation T, the image of a point X under the transformation
is given by the matrix equation TX  X ' . The matrix of T relates the coordinates of an
object X to the coordinates of its image X’ under the linear transformation. What matrix
relates the parameters of an object line L to its image L’ under the same linear
transformation.
2.
How many sets of object-image points are necessary to uniquely define a linear
transformation. Why?
3.
Demonstrate by example that the composition of linear transformations is not
commutative.
4.
Demonstrate by example that composition of linear transformations is associative.