the q form a basis for ?
Tonio is using a fundamental theorem about "bases".
A basis of vector space V has three properties:
1) They are independent.
2) They span the space.
3) The number of vectors in the basis is equal to the dimension of the space.
And if any two of those is true, the third is also true.
Tonio is assuming that you know that has dimension 3 so, since you have 3 vectors in your set (3) above is satisfied. It is only necessary to show that either (1) or (2) is satisfied and he is suggesting (1).
use coordinated vectors, express the first one as (note that the order was given in order to express those polynomials into coordinated vectors), then put each vector below of the other to get a matrix, compute its determinant, and it suffices to show it's not zero to have linear independence.