33. Let T: P₂→ P₂ be defined by T(ax² + bx + c) = ax^2 + (a-2b)x + b a. Determine whether p(x) = 2x^2 - 4x +6 is in the range of T. b. Find a basis for R(T).

T: P₂→ P₂ is a linearoperator. Determine whether the set\{T(1), T(x), T(x ^ 2)\} is a basis for P2. 19. . T(a * x ^ 2 + bx + c) = (a + b + c) * x ^ 2 + (a + b) * x + a

In each part, determine whether the given vector p(1) in P₂ belongs to span (p₁(t), p₂(t), p3(1)), where p_{1}(t) = t ^ 2 + 2t + 1 p_{2}(t) = t ^ 2 + 3 and p_{1}(t) = t - 1 (a) p(t) = t ^ 2 + t + 2 (b) p(t) = 2t ^ 2 + 2t + 3 (c) p(t) = - t ^ 2 + t - 4 (d) p(t) = - 2t ^ 2 + 3t + 1

Let Pn(R) = The vector space of polynomials with real coefficients and degree lessor equal to n. Show that the set {x + 1, x2 + x − 1, x2 − x + 1} is a basis for P2(R).Hence, determine the coordinates of the following elements: 2x−1, 1+x2, x2 +5x−1with respect to the above basis

Find the linear transformation T : ℝ2 → ℝ2 that has the values given below on the given basis.T 2 1 = 4 6,    T 1 1 = 3 7(a) First write a general vector v as a linear combination of the basis vectors.v =v1 v2 =  v1​−v2​ 2 1 +  2v2​−v1​ 1 1(b) This implies the following.T(v) = T v1 v2= 2v1​ v1​+6v2​ = v

Consider the following.x = 2 0−2, = 1 00, 0 10, 0 01, = 1 11, 0 11, 0 01 in ℝ3(a)Find the coordinate vectors [x]  and [x]  of x with respect to the bases and , respectively.[x]  =  [x]  =  (b)Find the change-of-basis matrix P  ← from to .P  ←  =  (c)Use your answer to part (b) to compute [x] , and compare your answer with the one found in part (a).[x]  =  (d)Find the change-of-basis matrix P  ← from to .P  ←  =  (e)Use your answers to parts (c) and (d) to compute [x] , and compare your answer with the one found in part (a).[x]  =