# Vector subspace question

• May 13th 2008, 12:11 PM
Jim Newt
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
• May 13th 2008, 12:20 PM
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?
• May 13th 2008, 12:23 PM
Jim Newt
Quote:

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?
• May 13th 2008, 01:23 PM
TheEmptySet
Quote:

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

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

I hope this helps.