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131
(ii)
Prove that
β
lies in the column space of
A
if and only if rank
(
[
A

β
]
)
=
rank
(
A
)
.
Solution.
Let
A
be Gaussian equivalent to an echelon matrix
U
,
so that there is a nonsingular matrix
P
with
PA
=
U
. Then
β
lies
in the row space
Ro
w(
A
)
if and only if
P
β
∈
U
)
.
(iii)
Does
β
=
(
0
,
−
3
,
5
)
lie in the subspace spanned by
α
1
=
(
0
,
−
2
,
3
)
,
α
2
=
(
0
,
−
4
,
6
)
,
α
3
=
(
1
,
1
,
−
1
)
?
Solution.
No.
4.30
(i)
Prove that an
n
×
n
matrix
A
over a
f
eld
k
is nonsingular if and
only if it is Gaussian equivalent to the identity
I
.
Solution.
Absent.
(ii)
Find the inverse of
A
=
231
−
110
101
.
Solution.
If
E
p
···
E
1
A
=
I
, then
A
−
1
=
E
−
1
1
E
−
1
p
. Con
clude that the elementary row operations which change
A
into
I
also change
I
into
A
−
1
. The answer is
A
−
1
=
1
4
1
−
3
−
1
11
−
1
−
13 5
.
4.31
(i)
Let
Ax
=
b
be an
m
×
n
linear system over a
f
eld
k
. Prove that
there exists a solution
x
=
(
x
1
,...,
x
n
)
with
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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