Determine if set is a vector space

[Laura1321412]

Homework Statement

Q) Determine if the set is a vector space under the given operations

The set of all pairs of real numbers of the form (1,x) with the operations (1,y)+(1,y')=(1,y+y') and k(1,y)=(1,ky)


2. Homework Equations / Solution Attempt

I know the axioms needed in this case, and I believe most all of them hold. The ones I am having trouble with in particular are

> There is an object 0 in V called the zero vector such that 0+u= u.
- But there isn't a zero vector if V is defined by (1,x) right?

> For each u in V there is an object -u in V, such that -u + u =0
- But -u would equal (1,-x) + (1,x) = (2,0) -- not 0


However, in the answer section of my book it says that this is a vector space under the given operations. I can't understand how the two above axioms hold... Any help is greatly appreciated!

 

[ystael]

> There is an object 0 in V called the zero vector such that 0+u= u.
- But there isn't a zero vector if V is defined by (1,x) right?

> For each u in V there is an object -u in V, such that -u + u =0
- But -u would equal (1,-x) + (1,x) = (2,0) -- not 0

First: There must be a zero vector, but the only property the zero vector needs to have is that $$0 + u = u + 0 = u$$ for every $$u \in V$$ -- using the definition of $$+$$ you are given. In other words, you shouldn't necessarily expect the zero vector to be $$(0, 0)$$. In fact, as you point out, $$(0, 0) \notin V$$.

Second: Be careful -- when you compute $$-u + u$$, what definition of $$+$$ should you be using?

 

[Fredrik]

You can (and should) use the report button to request that your other thread be deleted. (I also agree with what ystael said).