Chapter R.1, Problem 35PE

R denotes the field of real numbers, Q denotes the field of rationals, and Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures. Question 1 For each non-negative integer m, let R[x]m denote the vector space consisting of the polynomials in x with coefficients in R and of degree ≤ m. x²+2, V3 = 5. Prove that (V1, V2, V3) is a linearly independent (a) Let vi = x, V2 = list in R[x] 3. (b) Let V1, V2, V3 be as defined in (a). Find a vector v € R[×]3 such that (V1, V2, V3, V4) is a basis of R[x] 3. [8] [6] (c) Prove that the map ƒ from R[x] 2 to R[x]3 given by f(p(x)) = xp(x) — xp(0) is a linear map. [6] (d) Write down the matrix for the map ƒ defined in (c) with respect to the basis (2,2x + 1, x²) of R[x] 2 and the basis (1, x, x², x³) of R[x] 3. [5]

Question 4 (a) The following matrices represent linear maps on R² with respect to an orthonormal basis: = [1/√5 2/√5 [2/√5 -1/√5] " [1/√5 2/√5] A = B = [2/√5 1/√5] 1 C = D = = = [ 1/3/5 2/35] 1/√5 2/√5 -2/√5 1/√5' For each of the matrices A, B, C, D, state whether it represents a self-adjoint linear map, an orthogonal linear map, both, or neither. (b) For the quadratic form q(x, y, z) = y² + 2xy +2yz over R, write down a linear change of variables to u, v, w such that q in these terms is in canonical form for Sylvester's Law of Inertia. [6] [4]

part b please