14-6: Transformation Matrices

Unit 1 - Matrices

Learning Targets:• Find the images of points under different types of

transformations using matrices.

Here, we have Δ"#$ where:A = (0, 2)B = (3, 1)C = (2, -3)

We are going to transform Δ"#$ onto its image points %′(( + 2+, 3( + +).

A = (0, 2)B = (3, 1)C = (2, -3)

A’ = (4, 2)B‘= (5, 10)C’ = (-4, 3)

In this transformation, if the image of P(x, y) is P’(x’, y’), then:

! + 2$ = !&3! + $ = $&

This system can be written as the matrix equation:

1 23 1

!$ = !′

$′

It is customary to use the letter Tto represent both the transformation and its corresponding matrix.

! = 1 23 1 represents

!: ((, *) → (( + 2*, 3( + *)

Though we already have found the images of A, B, and C using the transformation rule, a more efficient method is to use matrix multiplication.

1 23 1 × 0 3 2

2 1 −3 = 4 5 −42 10 3

T A B C A’ B’ C’

Example:Consider the transformation !: #, % → (2#, −%) and figure F shown.

a) Find the transformation matrix T.

! = 2 00 −1

Example:Consider the transformation !: #, % → (2#, −%) and figure F shown.

b) Using matrices, find the images of the nine points determining F.

2 00 −1

1 2 22 1 3

3 4 43 3 4

4 5 52 1 3

= 2 4 4−2 −1 −3

6 8 8−3 −3 −4

8 10 10−2 −1 −3

Example:Consider the transformation !: #, % → (2#, −%) and figure F shown.

c) Plot the image points and draw F’. Compare the orientations of F and F’.

F and F’ have opposite orientations.

a) By counting squares, approximate the areas of F and F’ in the diagram given.

b) Find the ratio of the area of F’ to the area of F.

c) Find the determinant of T.

d) Compare your answers to parts b) and c)

Activity:

2:1

-2

They are opposite

Theorem 1:

Let ! = # $% & represent a linear transformation that maps a region R to an

image region R’. Then:

1. '()' *+ ,-'()' *+ , = absolute value of the determinant of T.

2. If the determinant of T is positive, then R’ and R have the same orientation.

If the determinant of T is negative, then R’ and R have the opposite orientations.

If the determinant of T is zero, then R’ lies on a line through the origin.

Theorem 2:

The columns of a transformation matrix are the images of the points (1, 0) and (0, 1), respectively.

If ! = # $% & , then

# $% &

10 = #

%# $% &

01 = $

&and

Example:Let S be a reflection in the y-axis. Find the matrix S.

This is what happens to the points (1, 0) and (0, 1) when each is reflected in the y-axis.

Image of 10 = −10

Image of 01 = 01

∴ & = −1 00 1 represents the

reflection in the y-axis.

Suppose !: # → #′ and &: #′ → #”. The single transformation that maps F to F” is called the composite of S and T, written as & ∘ !.

& ∘ ! # = &(! # )= & #′= #”

& ∘ !

&

!

Theorem 3:

Let S and T be transformations with corresponding matrices.

! = # $% & and ' = ( )

* ℎ

Then the matrix representing the transformations of ! ∘ ' is the matrix !', where

!' = # $% &

( )* ℎ

Example:

Reminder: ! = −1 00 1 and & = 2 0

0 −1

Consider the transformations given in Examples 1 & 2. Find the corresponding matrix for the transformation ! ∘ &.

So:

! ∘ & = !& = −1 00 1

2 00 −1 = −2 0

0 −1

Practice Problems:

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