show that the countable collection abcd a

show that the countable collection ( (a,b)*(c,d) | a<b and c<d and a,b,c,d are rational ) is a basis for R2.

Solution

You could explicitly solve m(a, b) + n(c, d) = (x,y) for any vector (x,y) in R2 and show it has a solution.

m*a + n*c = x
m*b + n*d = y

Clearly it has a solution,

So, They form a basis of R².