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F10 Homework 8.
Section 4.1.
4.
A problem which is almost the same is posted in the Course Documents area and
was brieﬂy shown in class.
You are actually expected to check all the 8 conditions.
6.
The set of non singular matrices does not contain the
O
(zero) matrix which is
singular. Accordingly this subset cannot be a subspace. (All subspaces contain the zero
vector).
Problem 11.
a) Let
A
x
1
=
b
and
A
x
2
=
b
, then
A
(
x
1
+
x
2
) =
A
x
1
+
A
x
2
= 2
b
.
x
1
+
x
2
is a
solution of
A
x
=
b
if and only if 2
b
=
b
, that is if and only if
b
=
0
.
Just the same,
A
(
c
x
1
) =
cA
x
1
=
c
b
and
c
x
1
is a solution of
A
x
=
b
if and only if
c
b
=
b
. Either
c
= 0 or not, this is possible if and only if
b
=
0
.
b) Since
V ⊂
R
n
, the addition and scalar multiplication in
V
enjoy the properties that
have been proved to hold in
R
n
.
c) For (3) to make sense, the vector
0
must be a solution. But
A
0
=
0
, so
0
∈ V
only
if
b
=
0
.
d) If (3) does not hold,
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This note was uploaded on 12/05/2011 for the course M 341 taught by Professor Hietmann during the Spring '08 term at University of Texas at Austin.
 Spring '08
 HIETMANN

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