LinearAlgebraNIII

math 111 assignment 4 fall 2014 2

T

4. This is a multipart graphical problem. You should print the following five pages (parts a – f)

and use them to present your solution.

In the diagram below, the 2×2 shaded square centred at the origin is bounded by the lines x = ±1 and y =

±1. Let T denote the transformation that maps the square into the 1×2 rectangle centred at (–1, 1.5).

(a) Use the structure theorem for affine transformations to find a matrix A and a vector b for which

T(x) = Ax + b.

[Check your work carefully with a suitable point, e.g. x = [1, 1], as you will be using this formula later.]

math 111 assignment 4 fall 2014 3

T

D

R

F

L

 

  



y

x

(b) Use the graphs at the right to show how T can be

decomposed as a composition of four basic transformations in

the following order: first a dilation D (using positive numbers),

followed by a rotation R through 90° (counter-clockwise),

followed by a reflection F, followed by a translation L. At the

very right, provide an expression for a general point  

  



y

x as it

passes through each of these steps so that at the end you get the

formula for T(x) you found in (a).

math 111 assignment 4 fall 2014 4

T

(c) This question (and part (d) below) are about TT  , the composite of T with itself. Find a formula for

TT  (x) = T(T(x)) of the form

T(T(x)) = Cx + d

 

  

 



  

  

  

 

 





 





 





 



  

  



f

e

y

x

dc

ba

y

x TT

Do this working not with the diagram, but with the formula T(x) = Ax + b that you derived in (a).

[Hint: to keep your notation simple, work first with the vector form T(x) = Ax + b finding expressions for

C and d in terms of A and b, then calculate C and d.]

math 111 assignment 4 fall 2014 5

T

(d) Now solve the problem of (c) graphically. To do this, start with the T-image of the central square

(that’s the 2×1 rectangle that came from the first application of T) and track it through D, R, F and L to

get the result of TT  . [You can check that you get the same result as (c)]. It turns out that you can nicely do the tracking for this on one diagram, drawing the resulting rectangle after each of the four steps.

Be sure you include the black shading.

PROVIDE THE IMAGE OF THE RECTANGLE

AFTER EACH OF THE FOUR STEPS

DRAW ONLY THE FINAL IMAGE:

THE IMAGE OF THE ORIGINAL CENTRAL

RECTANGLE UNDER TT 

math 111 assignment 4 fall 2014 6

invL

invF

invR

invD

 

  



y

x

T

(e) This problem asks you to find the inverse of T. Your job

now is to draw a shaded rectangle with the property that if you

apply T to it you will get the original 2×2 square. Use the

following geometric strategy: start with the original 2×2 square

and apply the inverses of component transformations D, R, F

and L in reverse order. As in part (b), provide an expression for

a general point  

  



y

x as it passes through each of these steps so

that at the end you get the formula for the inverse of T(x).

(f) Finally use algebra to check that you have found the inverse of

T. That is, take the general expression you found in (a) and apply

it to your final formula that you got at the bottom of this page and

check that you get  

  



y

x again. To help you keep track of your x’s

and y’s, it’s clearer if you use new variables, say  

  



v

u to name

your final answer below.