Question 10 (6 marks) Consider the inner product on P2 defined by rl 0 Find two...
Extra Credit: For the following two extra credit problems, look at the file "Notes on inner product spaces." Which will be posted on Blackboard on Tuesday 6/25 10. Consider the space P (degree 2 polynomials) with inner product f. 9) f()g(t)dt. Extra Credit: For the following two extra credit problems, look at the file "Notes on inner product spaces." Which will be posted on Blackboard on Tuesday 6/25 10. Consider the space P2 (degree 2 polynomials) with inner product S(t)g(t)dt....
[8 marks] For a function space, the scalar (or inner) product of two functions f(r) and 8() is defined as (.8) = f()8(r)dr (a) Show that this definition of the scalar product satisfies all axioms of an inner prod- uct. Brief answers are sufficient. (b) Consider the functions Lo(r) =1 and L(r) =r and L2(r) =-. You may assume that Lo, L1 and L2 are an orthogonal function set, with respect to the scalar product defined above. Consider an arbitrary...
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,...
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...
7 Consider the inner product space Co. 11 with the inner product defined by < 2,9 >= ( ( (x) g(x) dx (a) Show that f(x) = 1 and g(x) = 2x - 1 are orthogonal (b) Find ||g(2)|| (e) Find the distance d(f(x), g(x)) between f(x) and g(x)
Question 4) (6 points) Below are two unrelated questions that both deal with inner products. Note part (a) does not connect to part (b) (a) Consider the inner product on PX(R) defined by < p(x), g(x) >= } p(x)q(z)dir Show that the vectors x2 and 42 - 3 are orthogonal in this inner product space. (b) Conisder the vector space M.(R). Give an example to show why the following is not a valid inner product on this space: <A, B...
6. (15 pts) Consider an inner product on the vector space P2[-1, 1] of polynomials of degree 2 or less in the closed interval [-1, 1], defined as follows: (f, 9) = | f(t)g(t) dt, for all f, ge P2[-1, 1]. Apply the Gram-Schmidt process to the basis {3, t – 2,t2 + 1} to obtain an {x1, X2, X3} = %3D orthonormal basis.
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,...
Consider the inner product space V = P2(R) with (5,9) = { $(0)g(t) dt, and let T:VV be the linear operator defined by T(f) = x f'(x) +2f (x) +1. (i) Compute T*(1 + x + x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors ß for which [T]k is diagonal. If such a basis exists, find one.
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