2. Let ro < 1<..< n be n + 1 distinct points in IR. Define polynomials...
Let f (x) be a monic polynomial of degree n with distinct zeros ai,..., an. Prove -1 Let f (x) be a monic polynomial of degree n with distinct zeros ai,..., an. Prove -1
Q 1 Let D: P.(R) - P.(R) be the differentiation map Dp = p.Write down the n+1x n+1- matrix Mp of D relative to the usual ordered basis (P.. . Pr). Let C: P.(R) + R"+l be the isomorphism which sends polynomials to their (column) vector of coordinates with respect to the ordered basis (Po....Pn). Show that the column space of My is precisely C(Im(D)). More generally. CoD Mpoc as maps from P.(R) R+!
(1 point) Let P, be the vector space of all polynomials of degree 2 or less, and let 7 be the subspace spanned by 43x - 32x' +26, 102° - 13x -- 7 and 20.x - 15c" +12 a. The dimension of the subspace His b. Is {43. - 32" +26, 10x - 13.-7,20z - 150 +12) a basis for P2? choose ✓ Be sure you can explain and justify your answer. c. Abasis for the subspace His { }....
n If f(x) = Σ a;x' is a polynomial in R[x], recall the derivative f'(x) is a polynomial as well i=0 (we'll talk more about the fact that derivatives are linear, in chapter 3). Recall I write R[x]n for the polynomials of degree < N. Let P(x) = aixº be degree N, N i=0 a.k.a. assume an # 0. Show that the derivatives P(x), P'(x), ...,P(N)(x) form a basis of R[x]n (where p(N) means the Nth derivative of P).
please answer question 2 only, question1 is the information that might need for question 2 2. Define the divided difference f[xo,xi,'. . ,Tk] as the coefficient of rk in p in Q.1. Prove the following recurrence formula: f(ax1, 2,,X- f{X0, X1,**.,&k-1 f[xo, ,,Xk] 1. Let f a, b -» IR and ro, x1, , Tk be k + 1 distinct points in [a, b]. Show that there exists a unique polynomial pk of degree <k such that ph (xj)f(x), j...
2. Consider the polynomials 0-k (z) := (1 + z) for k-0,..., 10 and let B-bo,b1bo) can be shown that B is a basis for Pio the vector space of polynomials of degree at most 10. (You do not need to prove this.) Let Pk (z)-rk for k = 0, 1, . . . , 10, so that S = {po, pi, . . . , pio) is the standard basis for P10. Use Mathematica to: (a) Compute the change...
Hint: Apply the rank-nullity theorem to the linear map Pn → Rn+1 that sends p ?→ (p(x0), . . . , p(xn)). Then use the fact that if polynomial of degree ≤ n has n + 1 distinct roots, then it is the zero polynomial. (3 points) Application: polynomial interpolation. Let (20; yo), ..., (In; Yn) be n +1 points R2 with distinct x-coordinates. Show that there exists a unique polynomial p(t) of degree <n such that p(xi) = yi...
6. (a) Let V be a vector space over the scalars F, and let B = (01.62, ..., On) CV be a basis of V. For v € V, state the definition of the coordinate vector [v]s of v with respect to the basis B. [2 marks] (b) Let V = R$[x] = {ao + a11 + a222 + a3r | 20, 41, 42, 43 € R} the vector space of real polynomials of degree at most three. Write down...
Q6 (4+3+3+ 6=16 marks) Let Xo, X1, X2 be three distinct real numbers. For polynomials p(x) and q(x), define < p(x),q(x) >= p(xo)q(x0) + p(x1)q(x1) + p(x2)q(22). Let p(n) denote the vector space of all polynomials with degree more no than n. (i) Show that < .. > is an inner product in P(2). (ii) Is < ... > an inner product in P(3)? Explain why. (iii) Is <,:> an inner product in P(1)? Explain why. (iv) Consider Xo =...
row reduction in uncountable dimension. Part 2. (Row-reduction in countably-infinite dimension) Let V denote the vector space of polynomials (of all degrees). Recall that V is an infinite-dimensional vector space, but it has a countable basis. Consider Te Hom(V, V) defined as T(p())5p () 10p(x - 1) 2.1. Write T as an oo x oo matrix, in the standard basis 1,X, x2, 13,... of V 2.2. Write T as an oo x oo matrix, in the basis 1, + 1,...