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I find this very hard to understand. Ithinkyou mean that you want to determine whether or not the subset of the vector space of all polynomials of degree two or less, with the property that p'(1)= p(2), is a subspace.

To show that it is a subspace, you must show that, if p1 and p2 both satisfy that equation, then so does ap1+ bp2 for a, b any numbers.

(ap1'+ bp2')(1)= ap1'(1)+ bp2'(1)= a(p1(2))+ b(p2(2)).

To find a basis, remember that any polynomial of degree 2 or less can be written as . Then so "p'(1)= p(2)" becomes 2a+ b= 4a+ 2b+ c or 2a+ b+ c= 0.We can choose two of those to be whatever we choose and solve for the third. For example, if we take a= 1, b= 0 we get 2+ c= 0 or c= -2. is a basis vector. If we take a= 0, b= 1 we get 1+ c= 0 or c= -1. x- 1 is a We have no restriction on a and, whatever b is, c must be -b. x- 1 is a basis vector.

Those are the choices I would make and the basis I would determine. If we choose a= -1, b= 0, then c= 2 so is a basis vector. If we choose a= 0, b= -1, then c= 1 so is a basis vector. That is the basis your textbook gives.