Determining if certain sets are vector spaces

1. Dec 13, 2011

csc2iffy

1. The problem statement, all variables and given/known data

The set of all pairs of real numbers of the form (1,x) with the operations:
(1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx) k being a scalar

Is this a vector space?

2. Relevant equations
(1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx)

3. The attempt at a solution
I verified most of the axioms hold, but I'm unsure about the additive identity.
Can it be something other than the zero vector?
My attempt, "O" being the additive identity
O=(1,0)
A+O=(1,x)+(1,0)=(1,x)
This means the additive inverse must equal (1,0)
A+(-A)=(1,x)+(-1,-x)=(1,x+(-x))=(1,0)
If this isn't right, then I know it doesn't hold. I'm just a little confused. Thanks for any help

2. Dec 13, 2011

Dick

That's all correct. You understand this very well. They defined the operations in a way that was maybe intended to confuse you. But you weren't confused. (1,0) is the 'zero' vector.

3. Dec 13, 2011

Office_Shredder

Staff Emeritus
If A=(1,x) then -A is (1,-x) not (-1,-x). Otherwise it looks like you've got it. You can imagine just chopping off the 1 here and describing the element (1,x) by just the number x. Then what you really have is just the standard real numbers

4. Dec 13, 2011

Dick

Ooops. Missed that. Thanks for being careful.

5. Dec 13, 2011

SammyS

Staff Emeritus
Sort of OK. Besides the correction above, the wording needs a little work.

The sentence "This means the additive inverse must equal (1,0) ." should be changed to something like: "This means when the additive inverse of any number pair is added to that number pair, the result is (1,0)."

6. Dec 13, 2011

csc2iffy

Thank you all for the very quick responses!! That makes me feel so much better... I have a final coming up on this :)