Linear Algebra Subspaces

Hi,

I'm currently studying Linear Algebra, and we're in the section on subspaces. The only problem is that I'm having trouble showing closure under vector addition and scalar multiplication.

For example, I need to determine whether the given set is a subspace of R² (the x-y space).

What I am given is { (x1, x2)T | x1+x2 = 0} where T is for transpose, forming a column vector.

I believe that to show closure for vector addition, I need to show an example of (x, x)T + (y, y)T = 0 when y = -x, but I don't feel like this really proves what I need it to.

As for scalar multiplication, I haven't the slightest idea about where to begin.

Any pointers would be greatly appreciated! I've got another 40-some-odd problems of similar natures to do after this, but I figured I'd try and get the basics down first :)

Thanks!

So, to start, closure under addition:

let (a,b) and (a',b') be two members of our set (for convenience, I'm leaving off the T). What we want to show is that (a,b) + (a',b') must also be a member of the set.
To show that this is the case, we start by noting that
a + b = 0 and
a' + b' = 0

We then evaluate (a,b) + (a',b') as (a+a', b+b'), which is how vector addition is defined on R^2. We can then state that
(a+a') + (b+b') = (a+b) + (a'+b') = 0 + 0 = 0

thus, this new vector is a member of our set.

Now, scalar multiplication:
Take any (a,b) in the set, and take any scalar k. Multiplying, we get the vector
k*(a,b) = (ka, kb).
We then evaluate
ka + kb = k(a + b) = k*0 = 0
thus, this new vector is a member of our set.

Hope that helps