complex_numbers.20100115.4b5099ac2012a9.06205154 - Study...

Study sheet — Complex Numbers
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
Objective: To investigate the relationship between transformations of the plane and complex numbers.
Observation 1 Consider a transformation T that rotates objects in the plane by 30 ◦ and scales their size
by a factor of
5
3
. Suppose, for example, that the vector v makes an angle 45
◦
relative to the positive
horizontal axis of some coordinate system and has length 2 . Then, the result of applying the transfor-
mation T to the vector v is the vector T ( v ) that makes an angle 45
◦
+ 30
◦
= 75
◦
relative to the same
axis and has length 2 ·
5
3
=
10
3
.
The matrix of projections of the vector v onto the coordinate axis equals
μ
2 cos 45
◦
2 sin 45 ◦
¶
. (1)
Similarly, the matrix of projections of the vector T ( v ) onto the coordinate axes equals
μ
10
3
cos 75 ◦
10
3
sin 75
◦
¶
. (2)
But,
μ
10
3
cos 75 ◦
10
3
sin 75
◦
¶
=
10
3
μ
cos (30 ◦ + 45 ◦ )
sin (30
◦
+ 45
◦
)
¶
=
μ
5
3
· 2
¶ μ
cos 30
◦
cos 45
◦
− sin 30
◦
sin 45
◦
sin 30 ◦ cos 45 ◦ + cos 30 ◦ sin 45 ◦
¶
=
μ
5
3
cos 30
◦
−
5
3
sin 30
◦
5
3
sin 30 ◦
5
3
cos 30 ◦
¶ μ
2 cos 45
◦
2 sin 45 ◦
¶
, (3)
i.e., the matrix of projections of T ( v ) onto the coordinate axes is obtained by premultiplying the
matrix of projections of v onto the coordinate axes with the matrix
μ
5
3
cos 30 ◦ −
5
3
sin 30 ◦
5
3
sin 30
◦ 5
3
cos 30
◦
¶
. (4)
HHHHH
Observation 2 Consider a transformation T that rotates objects in the plane by 30 ◦ and scales their size
by a factor of
5
3
. Suppose, for example, that the vectors v and w make angles 30
◦
and − 30
◦
, respec-
tively, relative to the positive horizontal axis of some coordinate system and have lengths
1
2
and
3
2
,
1

respectively. Then, the results of applying the transformation T to the vectors v and w are the vectors
T ( v ) and T ( w ) that make angles 60
◦
and 0
◦
, respectively, relative to the same axis and have lengths
5
6
and
5
2
, respectively.
The vector sum v + w makes an angle ≈ − 16 . 10 ◦ relative to the positive horizontal axis of the co-
ordinate system and has length
√
13
2
. The vector sum T ( v ) + T ( w ) makes an angle ≈ 13 . 90 ◦ =
− 16 . 10
◦
+ 30
◦
relative to the same axis and has length
5
√
13
6
=
√
13
2
·
5
3
. It follows that, in this case,
T ( v ) + T ( w ) = T ( v + w ) . (5)
Theory: Denote by T
( r, θ )
a transformation that rotates objects in the plane by an angle θ and scales their
size by a factor r . Then, T
( r, θ )
is a linear transformation , i.e.,
T
( r, θ )
( α v + β w ) = α T
( r, θ )
( v ) + β T
( r, θ )
( w ) , (6)
where v and w are arbitrary vectors and α and β are arbitrary real numbers.