Determining if certain sets are vector spaces

[csc2iffy]

Homework Statement

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?

Homework Equations

(1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx)

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

 

[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.

 

[Office_Shredder]

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

 

[Dick]

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

Ooops. Missed that. Thanks for being careful.

 

[SammyS]

Homework Statement

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?

Homework Equations

(1,x)+(1,y)=(1,x+y) and k(1,x)=(1,kx)

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

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)."

 

[csc2iffy]

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