Graphics Reference
In-Depth Information
We tend to think about linear transformations as moving points around, but leaving
the origin fixed; we'll often use them that way. Equally important, however, is their
use in changing coordinate systems. If we have two coordinate systems on
R
2
with
the same origin, as in Figure 10.7, then every arrow has coordinates in both the
red and the blue systems. The two red coordinates can be written as a vector, as
u
r
can the two blue coordinates. The vector
u
, for instance, has coordinates
3
2
in
s
the red system and approximately
−
in the blue system.
0.2
3.6
Figure 10.7: Two different coor-
dinate systems for
R
2
; the vector
u
, expressed in the red coor-
dinate system, has coordinates
3
and
2
, indicated by the dot-
ted lines, while the coordinates
in the blue coordinate system
are approximately
−
0.2
and
3.6
,
where we've drawn, in each case,
the positive side of the first coor-
dinate axis in bold.
Inline Exercise 10.14:
Use a ruler to find the coordinates of
r
and
s
in each of
the two coordinate systems.
We could tabulate every imaginable arrow's coordinates in the red and blue
systems to convert from red to blue coordinates. But there is a far simpler way to
achieve the same result. The conversion from red coordinates to blue coordinates
is
linear
and can be expressed by a matrix transformation. In this example, the
matrix is
1
−
√
3
.
M
=
1
2
√
3
(10.34)
1
Multiplying
M
by the coordinates of
u
in the red system gets us
v
=
Mu
(10.35)
1
−
√
3
3
2
=
1
2
√
3
(10.36)
1
2
√
3
3
=
1
2
−
3
√
3
+
2
(10.37)
−
,
0.2
3.6
≈
(10.38)
which is the coordinate vector for
u
in the blue system.
Inline Exercise 10.15:
Confirm, for each of the other arrows in Figure 10.7,
that the same transformation converts red to blue coordinates.
By the way, when creating this example we computed
M
just as we did at the
start of the preceding section: We found the blue coordinates of each of the two
basis vectors for the red coordinate system, and used these as the columns of
M
.
In the special case where we want to go from the usual coordinates on a vector
to its coordinates in some coordinate system with basis vectors
u
1
,
u
2
, which are
unit vectors
and
mutually perpendicular,
the transformation matrix is one whose
rows
are the transposes of
u
1
and
u
2
.
(check for yourself that
these are unit length and perpendicular), then the vector
v
=
4
2
For example, if
u
1
=
3
and
u
2
=
−
/
5
4
/
5
4
/
5
3
/
5
, expressed in
u
-coordinates, is