MATH 102
SOLUTIONS TO HW #2
Section 2.1, problem 2.
The answers are as follows:
a) This
is
as subspace.
It consists of all vectors in
R
3
of the form (0
, b
2
, b
3
).
Notice that setting
b
2
=
b
3
= 0 gives the zero vector. Also, this set is closed under
addition because:
(0
, b
2
, b
3
) + (0
, c
2
, c
3
) = (0
, b
2
+
c
2
, b
3
+
b
3
). Likewise, we have
that
c
(0
, b
2
, b
3
) = (0
, cb
2
, cb
3
), which is of the correct general form.
b) This set
is not
a subspace. It consists of all vectors in
R
3
of the form (1
, b
2
, b
3
).
Notice that the vector 0 = (0
,
0
,
0) cannot be written in this form. It is not difficult
to see that the other vectorspace axioms (rules) are also violated for this set.
c) This set
is not
a subspace. It consists of all vectors which are either of the form
(
b
1
,
0
, b
3
) or (
b
1
, b
2
,
0). The problem is that if one adds two such vectors, say one of
each form, then the result will not necessarily be of the same form (i.e. like one or
the other). For example, let
x
= (1
,
0
,
1) and
y
= (1
,
1
,
0). Then
z
=
x
+
y
= (2
,
1
,
1)
which is not of the form: either (
b
1
,
0
, b
3
) or (
b
1
, b
2
,
0).
d) The set of all linear combinations of two vectors is
always
a subspace. For
example, in the case of this problem two such combinations may be added together
as follows:
(
a
1
(1
,
1
,
0)+
a
2
(2
,
0
,
1)
)
+
(
b
1
(1
,
1
,
0)+
b
2
(2
,
0
,
1)
)
= (
a
1
+
b
1
)(1
,
1
,
0)+(
a
2
+
b
2
)(2
,
0
,
1)
.
Therefore, we see that combinations are closed under addition. It is also clear that
the other axioms of a linear subspace are satisfied.
e) Solutions to a homogeneous linear equation are
always
a linear subspace. In
this case, let (
b
1
, b
2
, b
3
) and (
c
1
, c
2
, c
3
) be two solutions to the equation in the text.
Then a simple computation shows that their sum also satisfies the same equation:
(
b
3
+
c
3
)

(
b
2
+
c
2
) + 3(
b
1
+
c
1
) = (
b
3

b
2
+ 3
b
1
) + (
c
3

c
2
+ 3
c
1
) = 0 + 0 = 0
.
Similar computations show that the other two axioms of linear subspaces are also
satisfied in this case.
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 Fall '08
 SZYPOWSK
 Linear Algebra, Algebra, Vectors, Row

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