Hi,

**problem:**
Sometimes a subset of a vector space is itself a vector space (with respect to the linear operations already given). Consider, for example, the vector space

and the subsets

consisting of those vectors

for which

(a)

is real,

(b)

,

(c) either

,

(d)

,

(e)

In which of these cases is

a vector space?

**attempt:**
(a)

Not a vector space since multiplication by a scalar

yields

.

is not in

.

(b)

Is a vector space if

. If I think of this in terms of

, I would have the

plane if

I'm not sure if I should be thinking like this..

(c)

Again, if I think of it as

, I would get the

plane and the

plane.

This is not closed under addition,

,i.e., not a vector space.

I do not see which axiom for a vector space is violated though..

(d)

Is a vector space.

(e)

Hm, I would say no since it looks like it's not closed under addition.

Sorry for my pathetic attempts, can't do much better..

Thanks.