MA 35100 LECTURE NOTES: MONDAY, FEBRUARY 15
Inverse of a Linear Transformation
Consider an
n
×
m
matrix
A
, and the linear system
A ~x
=
~
y
. Explicitly,
a
11
x
1
+
a
12
x
2
+
· · ·
+
a
1
m
x
m
=
y
1
a
21
x
1
+
a
22
x
2
+
· · ·
+
a
2
m
x
m
=
y
2
.
.
.
.
.
.
.
.
.
=
.
.
.
a
n
1
x
1
+
a
n
2
x
2
+
· · ·
+
a
nm
x
m
=
y
n
Say that we can add and subtract suitable multiples of the equations in order to arrive at a system
in the form
x
1
=
b
11
y
1
+
b
12
y
2
+
· · ·
+
b
1
n
y
n
x
2
=
b
21
y
1
+
b
22
y
2
+
· · ·
+
b
2
n
y
n
.
.
.
=
.
.
.
.
.
.
.
.
.
x
m
=
b
m
1
y
1
+
b
m
2
y
2
+
· · ·
+
b
mn
y
n
This can be represented using matrices as
~x
=
B ~
y
for some
m
×
n
matrix
B
. If it is possible to
invert the system in this way, we say
A
is invertible, and denote
B
=
A

1
as its inverse.
We review a familiar definition which discusses a general phenomenon.
Definition.
A function
f
:
X
→
Y
is said to be
invertible
if for each
y
∈
Y
there
exists a unique
x
∈
X
such that
f
(
x
) =
y
.
If a function is invertible, we denote
f

1
:
Y
→
X
as that function which sends
y
=
f
(
x
) to the unique
x
.
Note that in general for an inverse to exist we need two statements: The equation
f
(
x
) =
y
has at
least one solution, and the equation
f
(
x
) =
y
has at most one solution.
We discuss a couple of examples. Consider the functions
R
→
R
defined by
f
(
x
) =
x
and
g
(
x
) =
x
2
.
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 Spring '10
 EGoins
 Linear Algebra, Algebra, Invertible matrix, Quantification

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