Theorem 53
(Thm 11.2, p.243)
.
Let A
= (
a
1
,
a
2
,
···
,
a
m
)
.
If
a
1
,
a
2
,
···
,
a
m
are linearly
independent if and only if
det
A
n
=
0
.
Proof.
a) “only if”: If
a
1
,
a
2
,
···
,
a
m
are linearly independent, then
A
is onetoone and
onto, hence has an inverse function. Call it
A

1
. Then
A

1
A
=
I
.
Hence det
A
n
=
0
.
b) “if”: If
a
1
,
a
2
,
···
,
a
m
are linearly dependent, then det
A
=
0
.
Theorem 54
(Variant of Thm 11.8, p.248)
.
The following statements are equivalent
(a) An m
×
m matrix A
= (
a
1
,
a
2
,
···
,
a
m
)
is a onetoone and onto linear function
from
R
m
to
R
m
.
(b) column vectors
a
1
,
a
2
,
···
,
a
m
are linearly independent.
(c)
det
A
n
=
0
.
(d) A is invertible.
Proof.
We only need to show that (c) implies (d). (c) implies that
A
is onetoone
and onto, hence that it has an inverse function. Call it
A

1
. Given
y
1
and
y
2
, let
y
1
=
A
x
1
and
y
2
=
A
x
2
. Then
s
y
1
+
r
y
2
=
A
(
s
x
1
+
r
x
2
)
, and hence