Recall that P₂(R) is the vector space of all polynomials of degree at most 2 with coefficients in R, with usual polynomial addition and scalar multiplication. For p = P₂ (R), let p' denote the first derivative of p with respect to x. Show that U = {p(x) = P₂(R) | p′(1) = p(0)} is a subspace of P₂(R).

Let V be the vector space of functions spanned by B = {b₁ = 2 sin x + 9 cos x, b₂ = 4 sin x + 7 cos x} where x C = {C₁ = sin x, C₂ = cos } where x of coordinates matrix P. C+B P = C←B [f(x)] c = [ The coordinates of a function f(x) relative to the basis B are [f(x)] B = [8] 6 coordinates of f(x) relative to C and find f(x). f(x) = Ex: 3 Ex: 3 nn 2 sin x + -, ne Z nn 2,n € Z is also a basis for V. Find the change COS x Find the

31. Let W be the subspace sp(sin 2x, cos 2x) of the vector space of all real-valued functions with domain R, and let B = (sin 2x, cos 2x). Find the matrix representation A for the linear transformation T: W→ W defined by T(f) = D²(f) + 2D(ƒ) + f.

Suppose a, b, and c are positive real numbers satisfying a2 = b2 + c2.r(t) = acos(t),bsin(t),csin(t) , 0 ≤ t ≤ 2πThen the vector-valued functiondescribes a tilted circle (a circle in a plane that is not parallel to one of the coordinateplanes). The center of the circle is O(0, 0, 0).(a) Show that |r(t)| is constant and determine the constant. This is the radius of thecircle.(b) Find an equation for the plane that contains the circle by doing the following:(i) Find three points on the circle.(ii) Use the three points to find a normal vector to the plane containing the three points.(iii) Find an equation for the plane. Check: does the plane contain the center of the circle?(iv) Simplify the equation as much as possible. (Since a, b and c are positive real numbers, you can divide by them without dividing by zero.)