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Math 20F Linear Algebra
Lecture 11: 4.3 Linearly independent sets: Basis.
2.3 Characterizations of Invertible Matrices.
Invertible Matrix Theorem (IMT)
Let
A
be a given
n
×
n
matrix. Then the following are equivalent:
a)
A
is invertible.
b)
A
is row equivalent to
I
.
c)
A
has
n
pivot positions
d) The equation
A
x
=
0
has only the trivial solution.
e) The columns of
A
are linearly independent.
f) The linear transformation
x
→
A
x
is onetoone.
g) The equations
A
x
=
b
has a solution for each
b
.
h) The columns of
A
span
R
n
.
i) The linear transformation
x
→
A
x
is onto.
j) There is an
n
×
n
matrix
C
such that
CA
=
I
.
k) There is an
n
×
n
matrix
D
such that
AD
=
I
.
l)
A
T
is invertible.
Basis
: Let
H
be a subspace of a vector space
V
.
{
b
1
,
···
,
b
n
}
is called a
basis
for
H
if
(i)
{
b
1
,
···
,
b
n
}
are linearly independent, and (ii)
{
b
1
,
···
,
b
n
}
span
H
.
Ex 3
Show that
e
1
=
1
0
0
,
e
2
=
0
1
0
,
e
3
=
0
0
1
is a basis for
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 Spring '03
 BUSS
 Algebra, Matrices, Sets

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