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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '03
 BUSS
 Linear Algebra, Algebra, Matrices, Sets, basis, ax, Row echelon form, Invertible Matrix Theorem

Click to edit the document details