Chapter 1
5.(b)Determine if is a subspace of
.
Solution: Since . Thus
is not a subspace of
.
5.(c)Determine if is a subspace of
.
Solution: Since at least one of
. Consider
and
. Both are in
but
. Thus,
is not a subspace of
.
5.(d)Determine if is a subspace of
.
Solution:
Clearly .
Consider and
both in
.
Then
Let and
Then .
is a subspace of
6. Give an example of
such that
is closed under addition and under taking additive inverses (meaning
whenever
), but
is not a subspace of
.
Solution 1:
Let
since
.
If then
since the sum of rationals is rational. Thus,
is closed under addition.
Also, are are rational since
are rational and the product of rationals is rational. Then
. So
is closed under taking additive inverses.
Now, let . Consider
. Then
since the product of an irrational and a non-zero rational is an irrational. Thus,
is not closed under scalar multiplication.
is not a subspace of
Solution 2: This is the one in the solution manual.
Let
since
.
Given and
then
and
and
.
Thus, is closed under addition.
Also, are integers since
are integers. Then
.
So is closed under taking additive inverses.
Consider and
.
Then .
Thus, is not closed under scalar multiplication.
is not a subspace of
7. Give an example of a nonempty subset of
such that
is closed under scalar multiplication, but
is not a subspace of
.
Solution:
Let
Just for fun, this is a different subset than the one found in the solution manual.
since
.
Let and
. If
or
, then
otherwise
. So
. Thus,
is closed under scalar multiplication.
Consider and
. Both are in
. But
.
is not a subspace of
.