Wednesday March 4
−
Lecture 23 :
The inverse of a linear transformation.
(Refers to
section 3.5)
Expectations:
1.
Recognize that an invertible matrix has RREF equal to
I
.
2.
Recognize that if
A
is the matrix induced by T and
A
is invertible then
T
is
invertible and
T
1
induces the matrix
A
1
.
3.
Use a rowreduction algorithm to find the inverse of
a matrix
A
.
23.0
Propostion – If
T
is a 1 – 1 linear transformation mapping
R
n
into
R
n
then
T
1
is
linear.
Proof: Suppose
y
1
and
y
2
belong to the image of
T
. such that
T
(
x
1
)
=
y
1
and
T
(
x
2
) =
y
2
.
Then
T
1
(
a
y
1
+
b
y
2
) =
T
1
(
a T
(
x
1
) +
b T
(
x
2
))
=
T
1
(
T
(
a
x
1
+
b
x
2
))
=
a
x
1
+
b
x
2
=
a T
1
(
y
1
) +
bT
1
(
y
2
)
So
T
1
is linear.
23.1
Theorem
−
Let
T
:
R
n
→
R
n
be a linear transformation. Let
A
be the standard matrix
induced by
T
.
Then
T
has an inverse
T
1
iff the matrix
A
is invertible. Furthermore, the
matrix induced by
T
1
is
A
1
.
Proof :
(
⇒
)
•
Suppose
T
is a linear transformation and
T
1
is its well defined inverse.
•
Let
A
be the standard matrix induced by
T
; let
B
be the standard matrix induced
by
T
1
.
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Then for all
x
in
R
n
,
x
=
T
1
(
T
(
x
)) =
B
(
A
(
x
))
and
x
=
T
(
T
1
(
x
)) =
A
(
B
(
x
)) . Thus
AB
(
x
) =
BA
(
x
) =
I
(
x
) for all
x
and so
AB = BA = I
. and so
B
=
A
1
.
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 Winter '08
 All
 Linear Algebra, Invertible matrix, Inverse element, ERO

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