== Let P3 have the inner product given by evaluation at -3, -1, 1, and 3....
4 2-5 Notice that these polynomials form an Let P3 have the inner product given by evaluation at -3, -1, 1, and 3. Let po(t) = 2, P (t) = 4t, and act) = orthogonal set with this inner product. Find the best approximation to p(t) = tº by polynomials in Span{Po-P1:9). The best approximation to p(t) = tº by polynomials in Span{Po.P7.93 is
Notice that these polynomials form an orthogonal set with this inner product. Find the best 1²-13 Let P2 have the inner product given by evaluation at -5, -1, 1, and 5. Let po(t) = 2, P1(t)=t, and q(t) = 12 approximation to p(t) = t by polynomials in Span{Po.P1,9}. The best approximation to p(t) = t by polynomials in Span{Po.P2,q} is
Let P3 have the inner product given by evaluation at -6, -1, 1, and 6. Let po(t) = 1, p1(t) = 2t, and P2 (t) = ? a. Compute the orthogonal projection of P2 onto the subspace spanned by Po and P4. b. Find a polynomial q that is orthogonal to po and p1, such that {PO,P1,93 is an orthogonal basis for Span{PO,P1.P2}. Scale the polynomial q so that its vector of values at ( - 6, - 1,1,6) is...
Let Ps have the inner product given by evaluation at -2, -1, 1, and 2. Let po(t)-1. P,()-t, and p20)- a. Compute the orthogonal projection of p2 onto the subspace spanned by Po and P1 b. Find a polynomial q that is orthogonal to Po and p,, such that Po P is an orthogonal basis for Span(Po P1, P2). Scale the polynomial q so that its vector of values at a2(Simplify your answer.) Let Ps have the inner product given...
Answer Question #12. Question #11 is only for reference 11. Let po, pi, and p2 be the orthogonal polynomials described in Example 5, where the inner product on P4 is given by evaluation at -2, -1, 0, 1, and 2. Find the orthogonal projection of tonto Span {po, pi, p2). 12. Find a polynomial p3 such that {po, p1, p2.p3} (see Exercise 11) is an orthogonal basis for the subspace P3 of P4. Scale the polynomial p3 so that its...
Let H={p() : p()= a + b + cf*: a,b,cer} (a)(3 marks) Show that H is a subspace of P3. (b) Let P1, P2, P3 be polynomials in H, such that Py(t) = 2, P2(t) = 1 +38P3(0)= -1-t-Use coordinate vectors in each of the following and justify your answer each part (1) (5 marks) Verify that {P1, P2, P3} form a linearly independent set in P3- (11) (2 marks) Verify that {P1, P2, P3} does not span P3. (111)...
Problem 1. Let the inner product (,) be defined by (u.v)xu (x)v (x) dx, and let the norm Iilbe defined by lIul-)Corhe target funtio), and work with the approximating space P4 Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials (x) through degree four. Standardize your polynomials such that p: (1) 1. (a) Form the five-by-five Gram matrix for this inner product with the basis functions p (x) degree 4 approximation o f (x) using the specified norm,...
let P3 denote the vector space of polynomials of degree 3 or less, with an inner product defined by 14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product defined by (p, q) Ji p(x)q(x) dr. Find an orthogo- nal basis for Ps that contains the vector 1+r. Find the norm (length) of each of your basis elements 14. Let Ps denote the vector space of polynomials of degree 3 or less,...
6. Let p;(xi = 0,... , n}, with degp;(x) = i, be a set of orthogonal polynomials with respect to the inner product f f(x)g(x) dx. Given a < b, let q(x) be the line mapping a to -1 and b to 1. Prove {p;(q(x))|i = 0,... , n} is a set of orthogonal polynomials with respect to the inner product f(x)g(x) dz, satisfying deg p;(q(x))= i - 6. Let p;(xi = 0,... , n}, with degp;(x) = i, be...
Problem 1. Let the inner prodct )be deined by (u.v)xu (x) v (x) dx, and let the norm |I-ll be defined by ull , ).Consider the target function f (x) with the approximating space P e', and work 2. Use Gram-Schmidt orthogonalization with this inner product to find orthogonal polynomials p (x) through degree four. Standardize your polynomials such that p, (1) 1 (b) Find the best degree 4 approximation to f(x) using the specified norm, and working with this...