1. ## Vector subspace question

Hi all,

Here's a question I'm struggling with:

Show that the set of all elements of R^2 of the form (a, -a), where a is any real number, is a subspace of R^2. Give a geometric interpretation of the subspace. Any suggestions?

Thanks,
Jim

2. 1) Does the set contain (0,0)?

2) Is the set closed under vector addition?

3) Is the set closed with respect to scalar multiplication?

3. Originally Posted by Plato
1) Does the set contain (0,0)?

2) Is the set closed under vector addition?

3) Is the set closed with respect to scalar multiplication?
How do I determine if the set is closed under vector addition or scalar multiplication?

4. Originally Posted by Jim Newt
How do I determine if the set is closed under vector addition or scalar multiplication?
Note: for all vectors the first and second component must be addative inverses.

let $\displaystyle v_1=(x,-x)$ and
$\displaystyle v_2=(y,-y)$ be any vectors in the subspace
Then
$\displaystyle v_1+v_2=(x,-x)+(y,-y)=(x+y,-x-y)=(x+y,-(x+y))$

Since the first and 2nd components are addative inverses the same is closed under vector addition.

I hope this helps.