Is R a Vector Space with Defined Operations? | Homework Statement

[amolv06]

Homework Statement

Let R denote the set of real numbers. Define scalar multiplication by \alpha x = \alpha x which is simply regular scalar multiplication, and vector addition is defined as x \oplus y = max(x,y). Is R a vector space with these operations?

Homework Equations

Some given above.

The Attempt at a Solution

There seems to be no zero vector to this equation as for any number k there exists another number k-1, so there is no single 0 vector for a vector space with the operations defined above. Is this reasoning correct?

 

[HallsofIvy]

Yes, that's true. So what is your answer to the question?

 

[amolv06]

Then it is not a proper vector space!

Thanks a lot.