(2)
S
◦
T
≠
T
◦
S
.
Theorem.
Let
S
:
R
2
→
R
2
and
T
:
R
2
→
R
2
be matrix transformations with matrices
A
and
B
respec
tively. Then
S
◦
T
is the matrix transformation with matrix
AB
.
Proof.
We have
S(v)
=
Av
and
T(v)
=
Bv
. Hence
[S
◦
T](v)
=
S(T(v))
=
S(Bv)
=
ABv.
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3.4.
MATRIX TRANSFORMATIONS OF
R
2
.
45
Example.
Let
R
θ
and
R
ψ
be rotations of the plane through the angles
θ
and
ψ
. Then obviously
R
θ
◦
R
ψ
=
R
θ
+
ψ
(and also
R
ψ
◦
R
θ
=
R
θ
+
ψ
).
There follows
cos
θ

sin
θ
sin
θ
cos
θ
cos
ψ

sin
ψ
sin
ψ
cos
ψ
=
A
θ
A
ψ
=
A
θ
+
ψ
=
cos
(θ
+
ψ)

sin
(θ
+
ψ)
sin
(θ
+
ψ)
cos
(θ
+
ψ)
,
which gives the standard trigonometric identities.
Example.
Let
S
be reflection in the line
y
=
x
and
T
be reflection in the
x
axis. We have
S(v)
=
Av
and
T(v)
=
Bv
, where
A
=
0
1
1
0
and
B
=
1
0
0

1
.
S
◦
T
is the matrix transformation induced by
AB
=
0
1
1
0
1
0
0

1
=
0

1
1
0
,
which is rotation
R
π/
2
through
π/
2
. The transformation
T
◦
S
is induced by
BA
=
1
0
0

1
0
1
1
0
=
0
1

1
0
,
which is rotation
R
3
π/
2
through
3
π/
2
.
3.4.2
Inverse of a Matrix Transformation.
Definition.
Let
T
:
R
2
→
R
2
be a matrix transformation induced by the matrix
A
.
T
is called
invertible
if
A
is invertible. In this case, the matrix transformation induced by
A

1
is called the inverse of
T
and
is denoted by
T

1
. That is,
T

1
is defined by
T

1
(v)
=
A

1
v,
v
∈
R
2
.
Observe that the composite
T
◦
T

1
is a matrix transformation induced by
AA

1
=
I
, and equally
T

1
◦
T
is a matrix transformation induced by
A

1
A
=
I
. That is,
[T
◦
T

1
](v)
=
v
and
[T

1
◦
T](v)
=
v
for all
v
∈
R
2
.
If
T
takes
v
into
w
=
T(v)
, then
T

1
takes
w
back into
v
.
Theorem.
Let
T
:
R
2
→
R
2
be a linear transformation induced by the matrix
A
. Then the following are
equivalent:
(1)
T
is invertible (i.e.
A
is invertible),
(2) There exists a linear transformation
S
:
R
2
→
R
2
such that
T
◦
S
=
I
R
2
and
S
◦
T
=
I
R
2
.
The matrix of
S
is
A

1
, so
S
=
T

1
.
Proof.
(
2
)
⇒
(
1
)
. Suppose
S
is induced by the matrix
A
. By the previous theorem,
AB
=
BA
=
I
, so
A
is invertible and
A

1
=
B
.
46
CHAPTER 3.
VECTOR GEOMETRY
Chapter 4
The Vector Space
R
n
.
Definition.
For
n
≥
1
, an ordered
n
tuple of real numbers
(a
1
, a
2
, . . . , a
n
)
is called an
n
vector. Written
in this form, it is called a
row vector
. Written as an
n
×
1
matrix
a
1
a
2
.
.
.
a
n
,
it is called a
column vector
. We will usually write our vectors as column vectors, and just call them
vectors. Let
R
n
be the set of all
n
vectors.
Notation.
Generally speaking, we won’t use arrows to denote vectors, except possibly if
n
=
2
or
3
, as
in the previous chapter.
Vectors are handled like matrices, in that if
X
=
x
1
x
2
.
.
.
x
n
and
Y
=
y
1
y
2
.
.
.
y
n
,
then
X
+
Y
=
x
1
+
y
1
x
2
+
y
2
.
.
.
x
n
+
y
n
and
cX
=
cx
1
cx
2
.