What Are the Zero and Unit Elements in R^2 for Proving Vector Space Axioms?

[indigojoker]

Let R^2 be a set containing all possible rows: (a b)

when using the 8 axioms to prove whether (a,b) is indeed a vector space, i have to show that there is a zero element and a unit element.

Is the zero element 0? or is it in matrix form such that W = (0 0) and W is contained in R^2?

Is the unit element 1? or is it in matrix form such that F=(1 1) and F is contained in R^2?

If I showed the 8 axioms are true, then does that show that R^2 is indeed a vector space?

 

[ircdan]

Your right about (0 0) and (1 1).
And yes just show the 8 axioms hold.

 

[HallsofIvy]

Your right about (0 0) and (1 1).
And yes just show the 8 axioms hold.

No, he's not right about (1, 1).

indigojoker, you shouldn't even have to think about that. The "zero element" acts like 0: x+ 0= 0 in the VECTOR addition. If you are adding vectors the 0 has to be a vector: (0, 0). On the other hand, scalar multiplication involves multiplying a scalar by a vector: in "1v= v", the "1" is a number, not a vector.