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Math
2940
Solutions, Fall
2011
Section
3
.
1
2
) Since the addition is usual, the ﬁrst four conditions are satisifed as they don’t involve the
scalar multiplication.
(5) fails as 1
·
(3
,
8) = (3
,
0)
6
= (3
,
8).
(6) holds as (
c
1
c
2
)(
x
1
, x
2
) = (
c
1
c
2
x
1
,
0) =
c
1
(
c
2
x
1
,
0) =
c
1
(
c
2
(
x
1
, x
2
))
.
(7) holds as
c
((
x
1
, x
2
) + (
y
1
+
y
2
)) =
c
(
x
1
+
y
1
, x
2
+
y
2
) = (
cx
1
+
cy
1
,
0) = (
cx
1
,
0) + (
cy
1
,
0) =
c
(
x
1
, x
2
) +
c
(
y
1
, y
2
).
(8) holds as (
c
1
+
c
2
)(
x
1
, x
2
) = ((
c
1
+
c
2
)
x
1
,
0) = (
c
1
x
1
+
c
2
x
1
,
0) = (
c
1
x
1
,
0) + (
c
2
, x
1
,
0) =
c
1
(
x
1
, x
2
) +
c
2
(
x
1
, x
2
).
4
) The ‘zero’ vector in the space of 2
×
2 matrices is
±
0 0
0 0
²
.
1
2
A
=
±
1

1
1

1
²
and

A
=
±

2 2

2 2
²
. The smallest subspace containing
A
consists of all scalar multiples of
A
,
namely all matrices of the from
±
2
r

2
r
2
r

2
r
²
as
r
runs through all real numbers.
10
) (a) Subspace
(b) Not a subspace  does not contain
~
0.
(c) Not a subspace  (0
,
1
,
1)
,
(1
,
0
,
0) are in the set, but their sum is not.
(d) Subspace
(e) Subspace
(f) Not a subspace  (1
,
2
,
3) is in the set, but it’s scalar multiple

1(1
,
2
,
3) = (

1
,

2
,

3)
is not in the set.
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