please solve all the parts 100% neat and clean thanks

Question 4 Let V denote the real vector space of polynomials of degree at most 4. Define an inner product on V as follows: For polynomials f(x), g(x) EV, F(x), g(x) = {\f(x)g(x) dx. (b) Let w1(x) = 1 + x4, wz(x) = 1 + x, w3(x) = x3, let B = { wi(x),w2(x), w3(x)}, and let W be the subspace of V spanned by B. (a) Show that B is a linearly independent set in V. (3 marks) Use the Gram-Schmidt orthogonalization procedure on the ordered basis B to find an orthogonal basis of W, and hence, determine an orthonormal basis D of W. (8 marks) Find a basis of w+ the orthogonal complement of W in V. (4 marks) (d) Define a linear operator T: WW be defined as follows: T(w1(x)) = w1(x) +w2(x) + w3(x) T(w2(x)) = w1(x) - w2(x) T(w3(x)) = w1(x) + wz(x) - 2w3(x) Compute [T]e, the matrix representation of T with respect to the orthogonal basis C found in part (b). (5 marks) Question 4 Let V denote the real vector space of polynomials of degree at most 4. Define an inner product on V as follows: For polynomials f(x), g(x) EV, F(x), g(x) = {\f(x)g(x) dx. (b) Let w1(x) = 1 + x4, wz(x) = 1 + x, w3(x) = x3, let B = { wi(x),w2(x), w3(x)}, and let W be the subspace of V spanned by B. (a) Show that B is a linearly independent set in V. (3 marks) Use the Gram-Schmidt orthogonalization procedure on the ordered basis B to find an orthogonal basis of W, and hence, determine an orthonormal basis D of W. (8 marks) Find a basis of w+ the orthogonal complement of W in V. (4 marks) (d) Define a linear operator T: WW be defined as follows: T(w1(x)) = w1(x) +w2(x) + w3(x) T(w2(x)) = w1(x) - w2(x) T(w3(x)) = w1(x) + wz(x) - 2w3(x) Compute [T]e, the matrix representation of T with respect to the orthogonal basis C found in part (b)