What you have written is very difficult to understand. Does "x2" mean x^{2}? I have no idea what "pi(x)= xi" means. My first guess would be p_{i}(x)= x^{i}so that p_{0}= 1, p_{1}= x, and p_{2}= x^{2}. Is that right?

If so then "the coordinate vector" is just something like (A, B, C) where A is the coefficient of 1, B is the coefficient of x, and C is the coefficient of x^{2}.

"Write each element of B as a linear combination of the elements of C" means to find a, b, c such that 1= a(2)+ b(3x- 2)+ c(2x^{2}- 3x+ 1), d, e, f, such that x= d(2)+ e(3x- 2)+ f(2x^{2}- 3x+ 1), and g, h, i, such that x^{2}= g(2)+ h(3x- 2)+ i(2x^{2}- 3x+ 1).