1. ## Subspace for R^3

{(x1, x2, x3)^T | x3 = x1 or x3 = x2}

I know it is not a subspace of R^3, however I'm not exactly sure why. I'm thinking it is because of the OR in the statement? In an earlier problem I had to prove that the same set and statement x1 = x2 = x3 is a subspace, which held true. But this OR is throwing me off. Any ideas?

2. Originally Posted by pakman
{(x1, x2, x3)^T | x3 = x1 or x3 = x2}

I know it is not a subspace of R^3, however I'm not exactly sure why. I'm thinking it is because of the OR in the statement? In an earlier problem I had to prove that the same set and statement x1 = x2 = x3 is a subspace, which held true. But this OR is throwing me off. Any ideas?
This is not a subspace it is not closed

Example

consider $\displaystyle v_1=(1,2,1) \mbox{ and } v_2=(-1,-2,-2)$

both $\displaystyle v_1,v_2$ fit the above criteria but

$\displaystyle v_1+v_2=(1,2,1)+(-1,-2,-2)=(0,0,-1)$

This is not in the "subspace" because $\displaystyle x_3=-1$

but both $\displaystyle x_1 \mbox{ and } x_2=0$