I have to show P1={a0 + a1x|a0 - 2a1 = 0} is a vector space using the 10 parts of the definition such as closed under addition and so on.

is the first statement right?

1. vector addition

Let a0+a1x and b0+b1x be two elements of P1. Then (a0+a1x)+(b0+b1x)=(a0+b0)+(a1+b1)x as long as (a0+b0)-2(a1+b1)=(a0-2a1)+(b0-2b1)=0+0=0.