Dr. Neal, WKU
MATH 307
Transformations in the Plane
Here are the matrix representations of some standard transformations in the xy -plane.
If desired, we can store these matrices in our calculator.
Reflection about y -axis
" !1 0%
'
T A ((x, y)) = (! x, y) so A = $
# 0 1&
Reflection about x -axis
"1 0 %
'
T B ((x, y)) = (x, ! y) so B = $
# 0 !1&
Reflection about origin
" !1 0 %
'
TC ((x, y)) = (! x, ! y) so C = $
# 0 !1&
Reflection about line y = x
! 0 1$
&
T D ((x, y)) = (y, x) so D = #
" 1 0%
Counterclockwise rotation by angle a
TE ((1, 0)) = (cos a, sin a) and TE ((0, 1)) = (! sin a, cosa) , so
" cosa ! sina%
'
E=$
# sina cosa &
Magnification by factor of constant k
! k 0$
&
TF ((1, 0)) = (k, 0) and TF ((0, 1)) = (0, k) so F = #
" 0 k%
To apply a transformation to a point (x,y) , we write the point as a column vector
! x$
# & and multiply on the left by the matrix transformation. For example, to reflect
" y%
" !3%
(!3, 4) about the y -axis, we first let G = $ ' , then
#4 &
" !1 0% " !3%
! 3$
' $ ' = # & = (3, 4) .
T A ((!3, 4)) = A ! G = $
# 0 1& # 4 &
" 4%
Thus, (3, 4) is the reflected point.
Get the Order Right!
To apply two transformations in a row, we multiply on the left by the first matrix, then
multiply again on the left by the next matrix.
!
Example 1. Consider the point (6, –2). First reflect it about the origin, then rotate it 45
counterclockwise.
Dr. Neal, WKU
" cos45!
In this case, E = $$
# sin45!
" 2 /2
= $$
# 2 /2
! sin 45! %' " 2 / 2
= $$
cos 45! '& # 2 / 2
! 2 / 2% " !1 0 % " 6 % " ! 2 / 2
'$
'!$ ' =$
2 / 2 '& # 0 !1& # !2& $# ! 2 / 2
"6 %
! 2 / 2%
'' . So we need ( E ! C ) ! $ '
# !2&
2 /2 &
2/2%
'
! 2 / 2'&
" 6 % " !4 2 % " !5.657%
'!$
$ ' = $$
'.
# !2& # !2 2 '& # !2.828&
!
Can we reverse the order? That is, can we first rotate it 45 , then reflect it about the
origin? Note that the matrix representation of the composition is
"! 2 / 2
"! 2 / 2
2/2%
2/2%
'' , which is the same as C ! E = $$
'' . So these
E ! C = $$
# ! 2 / 2 ! 2 / 2&
# ! 2 / 2 ! 2 / 2&
two transformations commute. For these two transformations, it doesn't matter which
one you apply first.
Example 2. Consider the point (6, –2). First reflect it about the y -axis, then rotate it
!
counterclockwise by 90 , then reflect it about the x -axis.
" cos90!
Here, E = $$
# sin90!
!sin 90! %' " 0 !1%
' . So to perform the transformations in correct
=$
cos90! '& # 1 0 &
# 6 & ! 2$
order, we use B ! E ! A ! % ( = # & .
$ "2' " 6%
!
Is this the same as rotating it first 90 , then reflecting it about the y -axis, then
#6 &
" !2%
! 2$
reflecting it about the x -axis? Now we use B ! A ! E ! % ( = $ ' " # & ; so we see
$ "2'
# !6&
" 6%
that A and E do not commute. You also can check that E ! A " A ! E .
Example 3. What is the inverse of the original transformation B ! E ! A in Example 2?
Describe it in words.
Recall that we reverse the order when taking the inverse of a product: (UV)!1 =
!1
!1
V !1 U !1 . Also note that with these transformations in the plane, A = A and B = B .
!1
!1
!1
"1
Thus, ( B ! E ! A) "1 = A " E " B = A ! E ! B . For the inverse, we apply B first,
!1
then E , then A . That is to undo the original B ! E ! A , we first reflect back across x !
!1
axis ( B ), then rotate counterclockwise 270 (or clockwise by 90º) ( E ), then reflect back
across y -axis ( A ).
Dr. Neal, WKU
Exercises
Let A , B , . . . , F be as defined at the beginning of this file.
!
1. Let TE be a counterclockwise rotation by
and let TF be shrinkage by a factor of 3.
6
(a) What are the matrix representations E and F ?
(b) Give the matrix representations of the inverses of TE and TF . Explain the effects of
these inverses.
2. (a) What is the single matrix representation of the transformation obtained by doing
!
a counterclockwise rotation by
followed by a reflection about the y -axis?
6
(b) Determine if the transformations in Part (a) commute.
3. Find the matrix representation of ( TC ! T A ! T D ! T B )–1 and explain in words the
sequence of operations that this inverse performs.
!
4. Let T B be a reflection about the x -axis, let TE be a counterclockwise rotation by ,
3
and let TF be magnification by a factor of 4.
(a) What is the matrix representation of the transformation given by doing a reflection
!
about the x -axis followed by counterclockwise rotation by ?
3
(b) Determine if the transformations in Part (a) commute.
(c) Find the matrix representation of ( TE ! T B ! TF )–1 and explain in words the
sequence of operations that the inverse performs.