**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 .