Graphics Reference
In-Depth Information
and
2
. The inverse of the matrix for
T
3
is
1
(note the negative sign).
2
01
−
1
3
The associated transformation also shears parallel to the
x
-axis, but vectors in
the upper half-plane are moved to the
left,
which undoes the moving to the right
done by
T
3
.
For these first three it was fairly easy to guess the inverse matrices, because
we could understand how to invert the transformation. The inverse of the matrix
for
T
4
is
21
−
,
1
4
(10.16)
21
which we computed using a general rule for inverses of 2
×
2 matrices (the only
such rule worth memorizing):
ab
cd
−
1
d
.
1
−
b
=
(10.17)
−
c
a
ad
−
bc
Finally, for
T
5
, the matrix has no inverse; if it did, the function
T
5
would be
invertible: It would be possible to identify, for each point in the codomain, a single
point in the domain that's sent there. But we've already seen this isn't possible.
Inline Exercise 10.10:
Apply the formula from Equation 10.17 to the matrix
for
T
5
to attempt to compute its inverse. What goes wrong?
We've said that every linear transformation really is just multiplication by some
matrix, but how do we
find
that matrix? Suppose, for instance, that we'd like to find
a linear transformation to flip our house across the
y
-axis so that the house ends
up on the left side of the
y
-axis. (Perhaps you can guess the transformation that
does this, and the associated matrix, but we'll work through the problem directly.)
The key idea is this: If we know where the transformation sends
e
1
and
e
2
,we
know the matrix. Why? We know that the transformation must have the form
T
x
y
=
ab
cd
x
y
;
(10.18)
we just don't know the values of
a
,
b
,
c
, and
d
. Well,
T
(
e
1
)
is then
T
1
0
=
ab
cd
1
0
=
a
c
.
(10.19)
Similarly,
T
(
e
2
)
is the vector
b
d
. So knowing
T
(
e
1
)
and
T
(
e
2
)
tells us all the
matrix entries. Applying this to the problem of flipping the house, we know that
T
(
e
1
)=
e
1
, because we want a point on the positive
x
-axis to be sent to the
corresponding point on the negative
x
-axis, so
a
=
−
1 and
c
=
0. On the other
hand,
T
(
e
2
)=
e
2
, because every vector on the
y
-axis should be left untouched, so
b
=
0 and
d
=
1. Thus, the matrix for the house-flip transformation is just
−
−
.
10
01
(10.20)
