Find a basis {p(x), q(x)} for the vector space {f(x) = P₂[x] f'(9) = f(1)} where P₂[x] is the vector space of polynomials in x with degree You can enter polynomials using notation e.g., 5+3xx for 5 + 3x². at most 2. p(x)= , q(x) =

Find a basis {p(x), g(x)} for the vector space {f(x) = P₂(1) | ƒ'(−2) = f(1)} where P₂(x) is the vector space of polynomials in with degree less than or equal to 2. |p(x) = , q(x) =

Find a basis {p(r), q(x)} for the vector space {f(x) e P2[z] | f' (-8) = f(1)} where P2[T] is the vector space of polynomials in x with degree at most 2.

Let w,=(1,0,0,1) and w=(-1,0,0,1). Let W=(w | w w, = w-w, = 0}, where w-w, denotes the dot product of w, & w. (a) Show that W is a vector space (over the field R). (b) Find the basis of W. (c) Find the dimension of W. (d) Is the answer to part (c) unique?