Subspace of R2 involving absolute value

[Mdhiggenz]

Homework Statement

{(x₁,x₂)^T| |x₂|=|x₂|}

So my first thought is we would have to check for both cases (x1,x1) and (-x1,-x1)

a=(x1,x1)^T b=(v1,v1)^T


βa=(βx1,βx1)^T for the case where a<0 βa(-βx1,-βx1)^T

thus it is closed under scalar multiplication.

a+b=(x1+v1,x1+v1) for the case where a and b<0 [-(x1+v1,x1+v1)]^T
thus closed under addition.

I would think this would be a subset, however the book says this is false. Where did I go wrong?

Homework Equations

The Attempt at a Solution

 

[Dick]

You are basically ignoring the absolute value. Are (1,1) and (1,-1) in your subspace? How about the sum?

 

[Mdhiggenz]

No because a+b would give you (2,0) and x1 and x2 have to be equal?

 

[Dick]

No because a+b would give you (2,0) and x1 and x2 have to be equal?

No, because |2| is not equal |0|. You don't have to have x1=x2 to be in the subspace. x1=(-x2) also works. But (2,0) doesn't fit either one.

 

[SammyS]

Homework Statement

{(x₁,x₂)^T| |x₂|=|x₂|}

Did you really mean |x₂| = |x₂| ?

 

[Mdhiggenz]

Oops that was a typo it was suppose to be |x1|=|x2|.

Also Dick, I'm a bit confused on why you chose (1,1) (1,-1)

 

[Dick]

Oops that was a typo it was suppose to be |x1|=|x2|.

Also Dick, I'm a bit confused on why you chose (1,1) (1,-1)

I was just trying to find two vectors in your set which didn't sum to be something in the set. Choosing one that satisfies x1=x2 and another that satisfies x1=(-x2) will do it. Nothing special about the 1's.

 

[Mdhiggenz]

But they must be equal? for instance we could use (2,2) and (2,-2) or (-2,2) and (2,2)

 

[Dick]

But they must be equal? for instance we could use (2,2) and (2,-2) or (-2,2) and (2,2)

(1,1) and (3,-3) work fine too. Their sum isn't in the set either.